这是序列A054261

第个素数包含数是包含前素数作为子字符串的最低数。例如，数字是最低的数字，其中包含前3个素数作为子字符串，使其成为第3个素数包含数。$n$$n$$235$

这是微不足道弄清楚前四个遏制素数是，，和，但随后变得更有趣。由于下一个素数是11，所以下一个素数包含数不是，而是因为它被定义为具有该属性的最小数。$2$$23$$235$$2357$$235711$$112357$

但是，当您超过11时，真正的挑战就来了。下一个主要收容编号为。请注意，在此数字中，子字符串和 是重叠的。该数字也与数字重叠。$113257$1113313

很容易证明此序列在增加，因为下一个数字需要满足该数字之前的所有条件，并且还要有一个子字符串。但是，该序列并不严格增加，如n=10和结果所示n=11。

挑战

您的目标是找到尽可能多的主要收容编号。您的程序应以有序的方式输出它们，从2开始并向上。

规则

1. 允许您对素数进行硬编码。
2. 不允许您对主要的安全壳号（2唯一的例外）或任何使挑战变得微不足道的魔术数字进行硬编码。请友好一点。
3. 您可以使用任何您想要的语言。请包括命令列表，以使环境为执行代码做好准备。
4. 您可以自由使用CPU和GPU，也可以使用多线程。

计分

官方评分将来自我的笔记本电脑（戴尔XPS 9560）。您的目标是在5分钟内生成尽可能多的主要收容编号。

眼镜

• 2.8GHz英特尔酷睿i7-7700HQ（3.8GHz增强）4核，8线程。
• 16GB 2400MHz DDR4内存
• NVIDIA GTX 1050
• Linux Mint 18.3 64位

到目前为止找到的数字，以及最后加上的质数：

 1 =>                                                       2 (  2)
 2 =>                                                      23 (  3)
 3 =>                                                     235 (  5)
 4 =>                                                    2357 (  7)
 5 =>                                                  112357 ( 11)
 6 =>                                                  113257 ( 13)
 7 =>                                                 1131725 ( 17)
 8 =>                                               113171925 ( 19)
 9 =>                                              1131719235 ( 23)
10 =>                                            113171923295 ( 29)
11 =>                                            113171923295 ( 31)
12 =>                                           1131719237295 ( 37)
13 =>                                          11317237294195 ( 41)
14 =>                                        1131723294194375 ( 43)
15 =>                                      113172329419437475 ( 47)
16 =>                                     1131723294194347537 ( 53)
17 =>                                   113172329419434753759 ( 59)
18 =>                                  2311329417434753759619 ( 61)
19 =>                                231132941743475375961967 ( 67)
20 =>                               2311294134347175375961967 ( 71)
21 =>                              23112941343471735375961967 ( 73)
22 =>                             231129413434717353759619679 ( 79)
23 =>                           23112941343471735359619678379 ( 83)
24 =>                         2311294134347173535961967837989 ( 89)
25 =>                        23112941343471735359619678378979 ( 97)
26 =>                      2310112941343471735359619678378979 (101)
27 =>                    231010329411343471735359619678378979 (103)
28 =>                 101031071132329417343475359619678378979 (107)
29 =>              101031071091132329417343475359619678378979 (109)
30 =>              101031071091132329417343475359619678378979 (113)
31 =>           101031071091131272329417343475359619678378979 (127)
32 =>           101031071091131272329417343475359619678378979 (131)
33 =>         10103107109113127137232941734347535961967838979 (137)
34 =>      10103107109113127137139232941734347535961967838979 (139)
35 =>   10103107109113127137139149232941734347535961967838979 (149)
36 => 1010310710911312713713914923294151734347535961967838979 (151)

感谢Ardnauld，Ourous和japh扩展了此列表。

请注意，n = 10和n = 11是相同的数字，因为是包含所有数字的最低数字，但它也包含。$113171923295$$[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]$$31$

作为参考，您可以使用以下事实：我为生成上述列表而编写的原始Python脚本在大约6分钟内计算了前12个术语。

附加规则

在获得第一个结果后，我意识到，很有可能最终的结果可以得到相同的分数。如果出现平局，获胜者将是产生结果最短的时间。如果两个或多个答案同样迅速地产生结果，那将简直是一场并列的胜利。

最后说明

5分钟的运行时间仅用于确保公平得分。我很想知道我们是否可以进一步推销OEIS序列（现在它包含17个数字）。使用Ourous的代码，我生成了直到的所有数字n = 26，但是我计划让代码运行更长的时间。

计分板

1. Python 3 + Google或工具：169
2. 斯卡拉（Scala）：137（非官方）
3. Concorde TSP求解器：84（非官方）
4. C ++（GCC）+ x86汇编：62
5. 干净：25
6. JavaScript（Node.js）：24

Python 3 + Google OR-Tools，在295秒内获得169分（官方得分）

怎么运行的

丢弃其他素数中包含的多余素数后，绘制一个有向图，其有一个从每个素数到其每个后缀的边，距离为零，一个从每个素数到其每个前缀的边，其距离由加数的位数定义。我们从图的字典顺序上寻找最短的最短路径，该路径从空前缀开始，经过每个素数（但不一定经过每个前缀或后缀），然后以空后缀结束。

例如，这是最优路径ε→11→1→13→3→31→1→17→ε→19→ε→23→ε→29→ε→5→ε的边，n = 11到输出字符串113171923295。

与直接解决旅行商问题相比，请注意，通过通过这些额外的后缀/前缀节点间接连接素数，而不是直接将素数彼此直接连接，我们大大减少了需要考虑的边数。但是由于多余的节点不必精确遍历一次，因此这不再是TSP的实例。

我们使用Google OR-Tools的增量CP-SAT约束求解器，首先最小化路径的总长度，然后依次最小化每组添加的数字。我们仅使用局部约束来初始化模型：每个素数都在一个后缀之前并且在一个前缀之后，而每个后缀/前缀都在相同数量的素数之前和之后。结果模型可能包含不连续的循环；如果是这样，我们将动态添加其他连接约束，然后重新运行求解器。

码

import multiprocessing
from ortools.sat.python import cp_model


def superstring(strings):
    def gen_prefixes(s):
        for i in range(len(s)):
            a = s[:i]
            if a in affixes:
                yield a

    def gen_suffixes(s):
        for i in range(1, len(s) + 1):
            a = s[i:]
            if a in affixes:
                yield a

    def solve():
        def find_string(s):
            found_strings.add(s)
            for i in range(1, len(s) + 1):
                a = s[i:]
                if (
                    a in affixes
                    and a not in found_affixes
                    and solver.Value(suffix[s, a])
                ):
                    found_affixes.add(a)
                    q.append(a)
                    break

        def cut(skip):
            model.AddBoolOr(
                skip
                + [
                    suffix[s, a]
                    for s in found_strings
                    for a in gen_suffixes(s)
                    if a not in found_affixes
                ]
                + [
                    prefix[a, s]
                    for s in unused_strings
                    if s not in found_strings
                    for a in gen_prefixes(s)
                    if a in found_affixes
                ]
            )
            model.AddBoolOr(
                skip
                + [
                    suffix[s, a]
                    for s in unused_strings
                    if s not in found_strings
                    for a in gen_suffixes(s)
                    if a in found_affixes
                ]
                + [
                    prefix[a, s]
                    for s in found_strings
                    for a in gen_prefixes(s)
                    if a not in found_affixes
                ]
            )

        def search():
            while q:
                a = q.pop()
                for s in prefixed[a]:
                    if (
                        s in unused_strings
                        and s not in found_strings
                        and solver.Value(prefix[a, s])
                    ):
                        find_string(s)
            return not (unused_strings - found_strings)

        while True:
            if solver.Solve(model) != cp_model.OPTIMAL:
                raise RuntimeError("Solve failed")

            found_strings = set()
            found_affixes = set()
            if part is None:
                found_affixes.add("")
                q = [""]
            else:
                part_ix = solver.Value(part)
                p, next_affix, next_string = parts[part_ix]
                q = []
                find_string(next_string)
            if search():
                break

            if part is not None:
                if part_ix not in partb:
                    partb[part_ix] = model.NewBoolVar("partb%s_%s" % (step, part_ix))
                    model.Add(part == part_ix).OnlyEnforceIf(partb[part_ix])
                    model.Add(part != part_ix).OnlyEnforceIf(partb[part_ix].Not())
                cut([partb[part_ix].Not()])
                if last_string is None:
                    found_affixes.add(next_affix)
                else:
                    find_string(last_string)
                q.append(next_affix)
                if search():
                    continue

            cut([])

    solver = cp_model.CpSolver()
    solver.parameters.num_search_workers = 4
    affixes = {s[:i] for s in strings for i in range(len(s))} & {
        s[i:] for s in strings for i in range(1, len(s) + 1)
    }
    prefixed = {}
    for s in strings:
        for a in gen_prefixes(s):
            prefixed.setdefault(a, []).append(s)
    suffixed = {}
    for s in strings:
        for a in gen_suffixes(s):
            suffixed.setdefault(a, []).append(s)
    unused_strings = set(strings)
    last_string = None
    part = None

    model = cp_model.CpModel()
    prefix = {
        (a, s): model.NewBoolVar("prefix_%s_%s" % (a, s))
        for a in affixes
        for s in prefixed[a]
    }
    suffix = {
        (s, a): model.NewBoolVar("suffix_%s_%s" % (s, a))
        for a in affixes
        for s in suffixed[a]
    }
    for s in strings:
        model.Add(sum(prefix[a, s] for a in gen_prefixes(s)) == 1)
        model.Add(sum(suffix[s, a] for a in gen_suffixes(s)) == 1)
    for a in affixes:
        model.Add(
            sum(suffix[s, a] for s in suffixed[a])
            == sum(prefix[a, s] for s in prefixed[a])
        )

    length = sum(prefix[a, s] * (len(s) - len(a)) for a in affixes for s in prefixed[a])
    model.Minimize(length)
    solve()
    model.Add(length == solver.Value(length))

    out = ""
    for step in range(len(strings)):
        in_parts = set()
        parts = []
        for a in [""] if last_string is None else gen_suffixes(last_string):
            for s in prefixed[a]:
                if s in unused_strings and s not in in_parts:
                    in_parts.add(s)
                    parts.append((s[len(a) :], a, s))
        parts.sort()
        part = model.NewIntVar(0, len(parts) - 1, "part%s" % step)
        partb = {}
        for part_ix, (p, a, s) in enumerate(parts):
            if last_string is not None:
                model.Add(part != part_ix).OnlyEnforceIf(suffix[last_string, a].Not())
            model.Add(part != part_ix).OnlyEnforceIf(prefix[a, s].Not())
        model.Minimize(part)
        solve()
        part_ix = solver.Value(part)
        model.Add(part == part_ix)
        p, a, last_string = parts[part_ix]
        unused_strings.remove(last_string)
        out += p
    return out


def gen_primes():
    yield 2
    n = 3
    d = {}
    for p in gen_primes():
        p2 = p * p
        d[p2] = 2 * p
        while n <= p2:
            if n in d:
                q = d.pop(n)
                m = n + q
                while m in d:
                    m += q
                d[m] = q
            else:
                yield n
            n += 2


def gen_inputs():
    num_primes = 0
    strings = []

    for new_prime in gen_primes():
        num_primes += 1
        new_string = str(new_prime)
        strings = [s for s in strings if s not in new_string] + [new_string]
        yield strings


with multiprocessing.Pool() as pool:
    for i, out in enumerate(pool.imap(superstring, gen_inputs())):
        print(i + 1, out, flush=True)

结果

这是前 8个内核/ 16线程系统在1½天内计算的前1000个主要安全壳数。

C ++（GCC） + x86汇编，在259秒内获得_{32 36} 62的得分（官方）

到目前为止计算的结果。之后，我的计算机内存不足65。

1 2
2 23
3 235
4 2357
5 112357
6 113257
7 1131725
8 113171925
9 1131719235
10 113171923295
11 113171923295
12 1131719237295
13 11317237294195
14 1131723294194375
15 113172329419437475
16 1131723294194347537
17 113172329419434753759
18 2311329417434753759619
19 231132941743475375961967
20 2311294134347175375961967
21 23112941343471735375961967
22 231129413434717353759619679
23 23112941343471735359619678379
24 2311294134347173535961967837989
25 23112941343471735359619678378979
26 2310112941343471735359619678378979
27 231010329411343471735359619678378979
28 101031071132329417343475359619678378979
29 101031071091132329417343475359619678378979
30 101031071091132329417343475359619678378979
31 101031071091131272329417343475359619678378979
32 101031071091131272329417343475359619678378979
33 10103107109113127137232941734347535961967838979
34 10103107109113127137139232941734347535961967838979
35 10103107109113127137139149232941734347535961967838979
36 1010310710911312713713914923294151734347535961967838979
37 1010310710911312713713914915157232941734347535961967838979
38 1010310710911312713713914915157163232941734347535961967838979
39 10103107109113127137139149151571631672329417343475359619798389
40 10103107109113127137139149151571631672329417343475359619798389
41 1010310710911312713713914915157163167173232941794347535961978389
42 101031071091131271371391491515716316717323294179434753596181978389
43 101031071091131271371391491515716316723294173434753596181917978389
44 101031071091131271371391491515716316717323294179434753596181919383897
45 10103107109113127137139149151571631671731792329418191934347535961978389
46 10103107109113127137139149151571631671731791819193232941974347535961998389
47 101031071091271313714915157163167173179181919321139232941974347535961998389
48 1010310710912713137149151571631671731791819193211392232941974347535961998389
49 1010310710912713137149151571631671731791819193211392232272941974347535961998389
50 10103107109127131371491515716316717317918191932113922322722941974347535961998389
51 101031071091271313714915157163167173179181919321139223322722941974347535961998389
52 101031071091271313714915157163167173179181919321139223322722923941974347535961998389
53 1010310710912713137149151571631671731791819193211392233227229239241974347535961998389
54 101031071091271313714915157163167173179211392233227229239241819193251974347535961998389
55 101031071091271313714915157163167173179211392233227229239241819193251972574347535961998389
56 101031071091271313714915157163167173179211392233227229239241819193251972572634347535961998389
57 101031071091271313714915157163167173179211392233227229239241819193251972572632694347535961998389
58 101031071091271313714915157163167173179211392233227229239241819193251972572632694347535961998389
59 1010310710912713137149151571631671731792113922332277229239241819193251972572632694347535961998389
60 101031071091271313714915157163167173211392233227722923924179251819193257263269281974347535961998389
61 1010310710912713137149151571631671732113922332277229239241792518191932572632692819728343475359619989
62 10103107109127131371491515716316717321139223322772293239241792518191932572632692819728343475359619989
63 1010307107109127131371491515716316717321139223322772293239241792518191932572632692819728343475359619989
64 10103071071091271311371391491515716316721173223322772293239241792518191932572632692819728343475359619989
65 10103071071091271311371491515716313916721173223322772293239241792518191932572632692819728343475359619989

这些都与基于Concorde的求解器的输出一致，因此它们很有可能是正确的。

变更日志：

• 所需上下文长度的计算错误。较早的版本1太大，并且还存在错误。得分：32 34

• 增加了对等境群体的优化。得分：34 36

• 改进了算法以正确使用上下文无关的字符串，以及其他一些优化方法。得分：36 62

• 添加了适当的记录。

• 添加了质数变体。

怎么运行的

警告：这是一个头脑风暴。如果只需要代码，请滚动到末尾。

缩略语：

• PCN：主要收容编号问题，即此挑战
• SCS：最短的通用超弦
• TSP：旅行业务员问题

该程序基本上使用针对TSP的教科书动态编程算法。

1. 加上从PCN / SCS（我们实际上正在解决的问题）到TSP的减少。
2. 加上使用项目上下文，而不是每个项目中的所有数字。
3. 再加上不能与其他素数的末端重叠的素数来细分问题。
4. 加上具有相同起始/结束位的素数的合并计算。
5. 加上预先计算的查找表和自定义哈希表。
6. 加上一些底层的预取和位打包。

那是很多潜在的错误。在尝试过anselm的条目并且无法从中诱骗任何错误的结果之后，我至少应该证明我的总体方法是正确的。

尽管基于Concorde的解决方案（快得多）快得多，但是它基于相同的简化，因此，此说明适用于两种解决方案。此外，该解决方案可适用于OEIS A054260（含质数的质数序列）；我不知道如何在TSP框架中有效解决该问题。因此它仍然有些相关。

减少TSP

让我们首先实际证明降低为TSP是正确的。我们有一组字符串，例如

A = 13, 31, 37, 113, 137, 211

我们想要找到包含这些项目的最小的超字符串。

知道长度就足够了

对于PCN，如果有多个最短的字符串，我们必须返回按字典顺序最小的字符串。但是，我们将研究一个不同的（更容易的）问题。

• SCS：给定一个初始前缀和一组项目，找到包含所有项目作为子字符串的最短字符串，并以该前缀开头。
• SCS-Length：只需找到SCS的长度即可。

如果能够求解SCS长度，则可以重构最小的解并获得PCN。如果我们知道最小的解决方案以我们的前缀开头，我们尝试通过按字典顺序附加每个项目并再次求解长度来扩展它。当我们找到解长度相同的最小项时，我们知道这必须是最小解中的下一个项（为什么？），因此将其添加并递归其余项。这种解决方案的方法称为自我简化。

游览最大重叠图

假设我们开始手动解决上述示例的SCS。我们可能会：

• 删除13和37，因为它们已经是其他项的子字符串。137例如，任何包含的解决方案还必须包含13和37。
• 开始考虑组合113,137 → 1137，211,113 → 2113等等。

这实际上是正确的做法，但是为了完整起见，让我们对其进行证明。采取任何SCS解决方案；例如，最短的超级字符串A是

2113137

可以将其分解为以下所有项目的串联A：

211
 113
   31
    137

（我们忽略了多余的项目13, 37。）请注意：

1. 每个项目的开始和结束位置至少增加1。
2. 每个项目都与前一个项目尽可能地重叠。

我们将显示每个最短的超字符串都可以通过以下方式分解：

1. 对于每对相邻的项目x,y，y开始和结束的位置比位置迟x。如果不正确，则要么x是的子字符串，y反之亦然。但是我们已经删除了所有属于子字符串的项目，因此不会发生。

2. 假设序列中的相邻项具有低于最大重叠，例如21113代替2113。但这会使多余的部分1变得多余。后面的项目都不需要首字母1（如2 1 113），因为它早于出现113，并且后面出现的所有项目113都不能以前面的数字开头113（请参见第1点）。类似的参数可防止最后一个多余的字符1（如211 1 3）被之前的任何项目使用211。但是，根据定义，我们最短的超字符串不会有多余的数字，因此不会发生这种非最大的重叠。

通过这些属性，我们可以将任何SCS问题转换为TSP：

1. 删除所有其他项目的子字符串的项目。
2. 创建一个有向图，每个图有一个顶点。
3. 对于每对项x，y添加到的边缘x，y其权重是通过最大重叠添加y到所增加的额外符号的数量x。例如，我们将在权重为1的情况下添加一条从211到的边113，因为在处又2113增加了一个数字211。重复从y到的边缘x。
4. 为初始前缀添加一个顶点，并从其边缘到所有其他项。

从初始前缀开始，此图上的任何路径都对应于该路径上所有项目的最大重叠级联，并且该路径的总权重等于级联的字符串长度。因此，每次访问所有项目至少一次的最低重量的巡回演出对应于最短的超弦。

这就是从SCS（和SCS-Length）到TSP的减少。

动态规划算法

这是一种经典算法，但是我们将对其进行很多修改，因此这里提醒您。

（我将其写为SCS-Length而不是TSP的算法。它们本质上是等效的，但是当我们进行特定于SCS的优化时，SCS词汇表会有所帮助。）

调用输入项集A和给定前缀P。对于每一个k-元素的子集S的A，和每一个元素e的S，我们计算的最短字符串的长度与开始P，包含了所有的S，并与两端e。这涉及存储从值(S, e)到其SCS长度的表。

当我们到每个子集S，该表需要已经包含了结果S - {e}对所有e在S。由于表可能会变得很大，因此我计算所有k元素子集的结果，然后计算k+1等等。为此，我们只需要存储一次k和的结果k+1。这样可以将内存使用量减少大约1倍sqrt(|A|)。

更详细一点：我实际上是在计算项目之间的最大总重叠量，而不是计算最小的SCS长度。（要获得SCS长度，只需从项目长度的总和中减去总重叠即可。）使用重叠有助于进行以下一些优化。

[2.]项目上下文

一个方面是可以与以下项目重叠的项目中最长的后缀。如果我们的项目113,211,311，然后11是上下文211和311。（这也是的前缀上下文113，我们将在第[4.]部分中进行介绍。）

在上面的DP算法中，我们跟踪了以每个项目结尾的SCS解决方案，但实际上我们并不关心SCS结束于哪个项目。我们需要知道的是上下文。因此，例如，如果同一集合的两个SCS 以23和结尾，43则从一个继续的任何SCS也将为另一个服务。

这是一个重要的优化，因为非平凡的质数仅以数字结尾1 3 7 9。1,3,7,9实际上，四个个位数的上下文（加上空上下文）足以计算素数最大为的PCN 131。

[3.]上下文无关的项目

其他人已经指出，许多质数都以数字开头2,4,5,6,8，例如23,29,41,43...。这些都不能与先前的质数重叠（除2和之外5，质数不能以这些数字结尾；2并且5已被删除为冗余数）。在代码中，这些被称为上下文无关字符串。

如果我们的输入中包含上下文无关的项目，则每个SCS解决方案都可以分为多个块

<prefix>... 23... 29... 41... 43...

并且每个区块中的重叠部分均独立于其他区块。我们可以在不改变SCS长度的情况下，对具有相同上下文的块进行混洗或交换项目。

因此，我们只需要跟踪可能的上下文多集，每个块一个即可。

完整示例：对于小于100的素数，我们有11个与上下文无关的项目及其上下文：

23 29 41 43 47 53 59 61 67 83 89
 3  9  1  3  7  3  9  1  7  3  9

我们最初的多集上下文：

1 1 3 3 3 3 7 7 9 9 9

该代码将这些称为组合上下文或ccontext。然后，我们只需要考虑其余项目的子集：

11 13 17 19 31 37 71 73 79 97

[4.]上下文合并

一旦我们获得3位数或更多的质数，就会有更多的冗余：

 101 151 181 191 ...
 107 127 157 167 197 ...
 109 149 1009 ...

这些组共享相同的开始和结束上下文（通常，这取决于输入中的其他素数），因此在重叠其他项目时它们是无法区分的。我们只关心重叠，因此我们可以将这些相等上下文组中的素数视为无法区分的。现在我们的DP子集被压缩成多个子集

4 × 1_1
5 × 1_7
3 × 1_9

（这也是为什么求解程序将重叠长度最大化而不是将SCS长度最小化的原因：此优化保留了重叠长度。）

简介：高级优化

运行INFO调试输出将打印统计信息，例如

solve: N=43, N_search=26, ccontext_size=18, #contexts=7, #eq_context_groups=16

该特定行用于前62个素数的SCS长度，2至293。

• 删除多余的项目后，我们剩下的43个素数互不相同。
• 有7个唯一的上下文：1,3,7,11,13,27加上空字符串。
• 在43个素数中，有17个是与上下文无关的：43,47,53,59,61,89,211,223,227,229,241,251,257,263,269,281,283。这些和给定的前缀（在这种情况下为空字符串）构成了初始组合上下文的基础。
• 在其余的26个项目（N_search）中，有16个非平凡的同等上下文组。

通过利用这些结构，SCS-Length计算仅需要检查8498336 (multiset, ccontext)组合。简单的动态编程将采取43×2^43 > 3×10^14步骤，而强行进行排列将采取6×10^52步骤。该程序仍然需要多次运行SCS-Length来重建PCN解决方案，但这并不需要花费更长的时间。

[5.，6.]低级优化

SCS-Length求解器不执行字符串操作，而是使用项和上下文的索引。我还预先计算了每个上下文和项目对之间的重叠量。

该代码最初使用GCC unordered_map，它似乎是一个具有链表存储区和主要哈希大小（即昂贵的划分）的哈希表。所以我写了自己的带有线性探测和2的幂的哈希表。这样可实现3倍加速和3倍内存减少。

每个表状态都由多个项目集，组合的上下文和重叠计数组成。它们打包成128位条目：8个用于重叠计数，56个用于多集（作为具有游程长度编码的位集），以及64个用于ccontext（以1分隔的RLE）。编码和解码ccontext是最棘手的部分，我最终使用了新PDEP指令（这是很新的东西，GCC尚无内在函数）。

最后，当散列表N很大时，访问散列表确实很慢，因为该表不再适合高速缓存。但是我们写入哈希表的唯一原因是为每个状态更新最知名的重叠计数。程序将此步骤拆分为一个预取队列，并且在实际更新该插槽之前，内部循环会预取每个表查找一些迭代。我计算机上的速度提高了2倍。

奖励：进一步的改进

AKA协和飞机有多快？

我对TSP算法了解不多，所以这里有一个大概的猜测。

Concorde使用分支剪切法来求解TSP。

• 将TSP编码为整数线性程序
• 它使用线性编程方法以及初始试探法来获得最佳游览距离的上下限
• 然后将这些界限输入分支和界限递归算法中，以寻找最佳解。如果计算的子树下限超过已知上限，则可以修剪搜索树的大部分内容
• 它还搜索切割平面以收紧LP松弛并获得更好的边界。通常，这些削减编码有关以下事实的知识：决策变量必须为整数

我们可以尝试一些显而易见的想法：

• 在SCS-Length求解器中进行修剪，尤其是在重建PCN解决方案时（此时，我们已经知道解决方案长度是多少）
• 得出一些易于计算的SCS下界，可用于帮助修剪
• 在质数分布中找到更多对称性或冗余以进行利用

但是，分支和剪切组合非常强大，因此对于大的值，我们可能无法击败最先进的求解器（如Concorde）N。

额外红利：主要遏制素数

与基于Concorde的解决方案不同，可以修改此程序以查找包含最小素数（OEIS A054260）的程序。这涉及三个更改：

1. $1/\ln(n)$

2. 修改SCS-Length求解器代码，以根据解决方案的数字总和是否可被3整除来对解决方案进行分类。这涉及向每个DP状态添加另一个条目，数字总和mod 3。这大大降低了主求解器陷入非素数排列的几率。这是我无法弄清楚如何转换为TSP的更改。可以使用ILP对其进行编码，但是随后我必须学习称为“子巡视不等式”的东西以及如何生成这些东西。

3. 可能所有最短的PCN都可被3整除。在这种情况下，最小素数容纳素数必须比PCN长至少一位。如果我们的SCS-Length求解器检测到此问题，则解决方案重建代码可以选择在过程中的任何一点添加一个额外的数字。它尝试0..9像以前一样按字典顺序将每个可能的数字和每个剩余的项目添加到当前解决方案前缀中。

通过这些更改，我可以获得直至的解决方案N=62。除了47，重构代码被卡住并在一百万步之后放弃（我不知道为什么）。主要遏制素数是：

1 2
2 23
3 523
4 2357
5 112573
6 511327
7 1135217
8 1113251719
9 11171323519
10 113171952923
11 113171952923
12 11131951723729
13 11317237419529
14 1131723294375419
15 113172329541947437
16 1131723294195343747
17 1113172329419434753759
18 11231329417437475361959
19 231132941743475375967619
20 2311294134347175967619537
21 23112941343471735967619537
22 231129413434717359537679619
23 23112941343471735375961983679
24 11231294134347173535961967983789
25 23112941343471735359679837619789
26 2310112941343471735359619783789679
27 231010329411343471735359619678379897
28 101031071132329417343475359619798376789
29 101031071091132329417343475359619767898379
30 101031071091132329417343475359619767898379
31 1010310710911131272329417343475359619678979837
32 1010310710911131272329417343475359619678979837
33 10103107109113127137232941734347535978961967983
34 10103107109113127137139232941734347535961967838979
35 10103107109113127137139149232941734347535961976798389
36 1010310710911312713713914923294151734347535976198389679
37 1010310710911312713713914915157232941734347535967619798389
38 10103107109111312713713914915157163232941734347535967897961983
39 10103107109113127137139149151571631672329417343475961979838953
40 10103107109113127137139149151571631672329417343475961979838953
41 10103107109111312713713914915157163167173232941794347535976198983
42 1010310710911131271371391491515716316717323294179434761819535989783
43 1010310710911131271371391491515716316723294173434753596181917989783
44 101031071091131271371391491515716316717323294179434753836181919389597
45 10103107109113127137139149151571631671731792329418191934347538961975983
46 101031071091113127137139149151571631671731791819193232941974347535989836199
47 (failed)
48 1010310710912713137149151571631671731791819193211392232941974347895359836199
49 10103107109112713137149151571631671731791819193211392232272941974347619983535989
50 10103107109127131371491515716316717317918191932113922322722941974347595389836199
51 101031071091271313714915157163167173179181919321139223322722941974347595389619983
52 101031071091271313714915157163167173179181919321139223322722923941974347538361995989
53 10103107109112713137149151571631671731791819193211392233227229239241974347619983538959
54 101031071091271313714915157163167173179211392233227229239241819193251974347619953835989
55 1010310710911271313714915157163167173179211392233227229239241819193251974325747596199538983
56 101031071091271313714915157163167173179211392233227229239241819193251972572634347619959895383
57 101031071091271313714915157163167173179211392233227229239241819193251972572632694359538983619947
58 101031071091271313714915157163167173179211392233227229239241819193251972572632694359538983619947
59 1010310710912713137149151571631671731792113922332277229239241819193251972572632694347535983896199
60 1010310710911271313714915157163167173211392233227722923924179251819193257263269281974347535961998389
61 1010310710912713137149151571631671732113922332277229239241792518191932572632692819728343538947619959
62 10103107109127131371491515716316717321139223322772293239241792518191932572632692819728343534759896199

码

编译

g++ -std=c++14 -O3 -march=native pcn.cpp -o pcn

对于素数版本，还与GMPlib链接，例如

g++ -std=c++14 -O3 -march=native pcn-prime.cpp -o pcn-prime -lgmp -lgmpxx

该程序使用PDEP指令，该指令仅在最新的（Haswell +）x86处理器上可用。我的计算机和maxb都支持它。如果您的计算机不支持，则该程序将以较慢的软件版本进行编译。发生这种情况时，将打印一个编译警告。

#include <cassert>
#include <cstdlib>
#include <cstring>
#include <iostream>
#include <vector>
#include <unordered_map>
#include <string>
#include <algorithm>
#include <array>

using namespace std;

void debug_dummy(...) {
}

#ifndef INFO
//#  define INFO(...) fprintf(stderr, __VA_ARGS__)
#  define INFO debug_dummy
#endif

#ifndef DEBUG
//#    define DEBUG(...) fprintf(stderr, __VA_ARGS__)
#  define DEBUG debug_dummy
#endif

bool is_prime(size_t n)
{
    for (size_t d = 2; d * d <= n; ++d) {
        if (n % d == 0) {
            return false;
        }
    }
    return true;
}

// bitset, works for up to 64 strings
using bitset_t = uint64_t;
const size_t bitset_bits = 64;

// Find position of n-th set bit of x
uint64_t bit_select(uint64_t x, size_t n) {
#ifdef __BMI2__
    // Bug: GCC doesn't seem to provide the _pdep_u64 intrinsic,
    // despite what its manual claims. Neither does Clang!
    //size_t r = _pdep_u64(ccontext_t(1) << new_context, ccontext1);
    size_t r;
    // NB: actual operand order is %2, %1 despite the intrinsic taking %1, %2
    asm ("pdep %2, %1, %0"
         : "=r" (r)
         : "r" (uint64_t(1) << n), "r" (x)
         );
    return __builtin_ctzll(r);
#else
#  warning "bit_select: no x86 BMI2 instruction set, falling back to slow code"
    size_t k = 0, m = 0;
    for (; m < 64; ++m) {
        if (x & (uint64_t(1) << m)) {
            if (k == n) {
                break;
            }
            ++k;
        }
    }
    return m;
#endif
}

#ifndef likely
#  define likely(x) __builtin_expect(x, 1)
#endif
#ifndef unlikely
#  define unlikely(x) __builtin_expect(x, 0)
#endif

// Return the shortest string that begins with a and ends with b
string join_strings(string a, string b) {
    for (size_t overlap = min(a.size(), b.size()); overlap > 0; --overlap) {
        if (a.substr(a.size() - overlap) == b.substr(0, overlap)) {
            return a + b.substr(overlap);
        }
    }
    return a + b;
}

vector <string> dedup_items(string context0, vector <string> items)
{
    vector <string> items2;
    for (size_t i = 0; i < items.size(); ++i) {
        bool dup = false;
        if (context0.find(items[i]) != string::npos) {
                dup = true;
        } else {
            for (size_t j = 0; j < items.size(); ++j) {
                if (items[i] == items[j]?
                    i > j
                        : items[j].find(items[i]) != string::npos) {
                    dup = true;
                    break;
                }
            }
        }
        if (!dup) {
            items2.push_back(items[i]);
        }
    }
    return items2;
}

// Table entry used in main solver
const size_t solver_max_item_set = bitset_bits - 8;
struct Solver_entry
{
    uint8_t score : 8;
    bitset_t items : solver_max_item_set;
    bitset_t context;

    Solver_entry()
    {
        score = 0xff;
        items = 0;
        context = 0;
    }
    bool is_empty() const {
        return score == 0xff;
    }
};

// Simple hash table to avoid stdlib overhead
struct Solver_table
{
    vector <Solver_entry> t;
    size_t t_bits;
    size_t size_;
    size_t num_probes_;

    Solver_table()
    {
        // 256 slots initially -- this needs to be not too small
        // so that the load factor formula in update_score works
        t_bits = 8;
        size_ = 0;
        num_probes_ = 0;
        resize(t_bits);
    }
    static size_t entry_hash(bitset_t items, bitset_t context)
    {
        uint64_t h = 0x3141592627182818ULL;
        // Add context first, since its bits are generally
        // less well distributed than items
        h += context;
        h ^= h >> 23;
        h *= 0x2127599bf4325c37ULL;
        h ^= h >> 47;
        h += items;
        h ^= h >> 23;
        h *= 0x2127599bf4325c37ULL;
        h ^= h >> 47;
        return h;
    }
    size_t probe_index(size_t hash) const {
        return hash & ((size_t(1) << t_bits) - 1);
    }
    void resize(size_t t2_bits)
    {
        assert (size_ < size_t(1) << t2_bits);
        vector <Solver_entry> t2(size_t(1) << t2_bits);
        for (auto entry: t) {
            if (!entry.is_empty()) {
                size_t h = entry_hash(entry.items, entry.context);
                size_t mask = (size_t(1) << t2_bits) - 1;
                size_t idx = h & mask;
                while (!t2[idx].is_empty()) {
                    idx = (idx + 1) & mask;
                    ++num_probes_;
                }
                t2[idx] = entry;
            }
        }
        t.swap(t2);
        t_bits = t2_bits;
    }
    uint8_t update_score(bitset_t items, bitset_t context, uint8_t score)
    {
        // Ensure we can insert a new item without resizing
        assert (size_ < t.size());

        size_t index = probe_index(entry_hash(items, context));
        size_t mask = (size_t(1) << t_bits) - 1;
        for (size_t p = 0; p < t.size(); ++p, index = (index + 1) & mask) {
            ++num_probes_;
            if (likely(t[index].items == items && t[index].context == context)) {
                t[index].score = max(t[index].score, score);
                return t[index].score;
            }
            if (t[index].is_empty()) {
                // add entry
                t[index].score = score;
                t[index].items = items;
                t[index].context = context;
                ++size_;
                // load factor 4/5 -- ideally 2-3 average probes per lookup
                if (5*size_ > 4*t.size()) {
                    resize(t_bits + 1);
                }
                return score;
            }
        }
        assert (false && "bug: hash table probe loop");
    }
    size_t size() const {
        return size_;
    }
    void swap(Solver_table table)
    {
        t.swap(table.t);
        ::swap(size_, table.size_);
        ::swap(t_bits, table.t_bits);
        ::swap(num_probes_, table.num_probes_);
    }
};

/*
 * Main solver code.
 */
struct Solver
{
    // Inputs
    vector <string> items;
    string context0;
    size_t context0_index;

    // Mapping between strings and indices
    vector <string> context_to_string;
    unordered_map <string, size_t> string_to_context;

    // Items that have context-free prefixes, i.e. prefixes that
    // never overlap with the end of other items nor context0
    vector <bool> contextfree;

    // Precomputed contexts (suffixes) for each item
    vector <size_t> item_context;
    // Precomputed updates: (context, string) to overlap amount
    vector <vector <size_t>> join_overlap;

    Solver(vector <string> items, string context0)
        :items(items), context0(context0)
    {
        items = dedup_items(context0, items);
        init_context_();
    }

    void init_context_()
    {
        /*
         * Generate all relevant item-item contexts.
         *
         * At this point, we know that no item is a substring of
         * another, nor of context0. This means that the only contexts
         * we need to care about, are those generated from maximal join
         * overlaps between any two items.
         *
         * Proof:
         * Suppose that the shortest containing string needs some other
         * kind of context. Maybe it depends on a context spanning
         * three or more items, say X,Y,Z. But if Z ends after Y and
         * interacts with X, then Y must be a substring of Z.
         * This cannot happen, because we removed all substrings.
         *
         * Alternatively, it depends on a non-maximal join overlap
         * between two strings, say X,Y. But if this overlap does not
         * interact with any other string, then we could maximise it
         * and get a shorter solution. If it does, then call this
         * other string Z. We would get the same contradiction as in
         * the previous case with X,Y,Z.
         */
        size_t N = items.size();
        vector <size_t> max_prefix_overlap(N), max_suffix_overlap(N);
        size_t context0_suffix_overlap = 0;
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                if (i == j) continue;
                string joined = join_strings(items[j], items[i]);
                size_t overlap = items[j].size() + items[i].size() - joined.size();
                string context = items[i].substr(0, overlap);
                max_prefix_overlap[i] = max(max_prefix_overlap[i], overlap);
                max_suffix_overlap[j] = max(max_suffix_overlap[j], overlap);

                if (string_to_context.find(context) == string_to_context.end()) {
                    string_to_context[context] = context_to_string.size();
                    context_to_string.push_back(context);
                }
            }

            // Context for initial join with context0
            {
                string joined = join_strings(context0, items[i]);
                size_t overlap = context0.size() + items[i].size() - joined.size();
                string context = items[i].substr(0, overlap);
                max_prefix_overlap[i] = max(max_prefix_overlap[i], overlap);
                context0_suffix_overlap = max(context0_suffix_overlap, overlap);

                if (string_to_context.find(context) == string_to_context.end()) {
                    string_to_context[context] = context_to_string.size();
                    context_to_string.push_back(context);
                }
            }
        }
        // Now compute all canonical trailing contexts
        context0_index = string_to_context[
                           context0.substr(context0.size() - context0_suffix_overlap)];
        item_context.resize(N);
        for (size_t i = 0; i < N; ++i) {
            item_context[i] = string_to_context[
                                items[i].substr(items[i].size() - max_suffix_overlap[i])];
        }

        // Now detect context-free items
        contextfree.resize(N);
        for (size_t i = 0; i < N; ++i) {
            contextfree[i] = (max_prefix_overlap[i] == 0);
            if (contextfree[i]) {
                DEBUG("  contextfree: %s\n", items[i].c_str());
            }
        }

        // Now compute all possible overlap amounts
        join_overlap.resize(context_to_string.size(), vector <size_t> (N));
        for (size_t c_index = 0; c_index < context_to_string.size(); ++c_index) {
            const string& context = context_to_string[c_index];
            for (size_t i = 0; i < N; ++i) {
                string joined = join_strings(context, items[i]);
                size_t overlap = context.size() + items[i].size() - joined.size();
                join_overlap[c_index][i] = overlap;
            }
        }
    }

    // Main solver.
    // Returns length of shortest string containing all items starting
    // from context0 (context0's length not included).
    size_t solve() const
    {
        size_t N = items.size();

        // Length, if joined without overlaps. We try to improve this by
        // finding overlaps in the main iteration
        size_t base_length = 0;
        for (auto s: items) {
            base_length += s.size();
        }

        // Now take non-context-free items. We will only need to search
        // over these items.
        vector <size_t> search_items;
        for (size_t i = 0; i < N; ++i) {
            if (!contextfree[i]) {
                search_items.push_back(i);
            }
        }
        size_t N_search = search_items.size();

        /*
         * Some groups of strings have the same context transitions.
         * For example "17", "107", "127", "167" all have an initial
         * context of "1" and a trailing context of "7", no other
         * overlaps are possible with other primes.
         *
         * We group these strings and treat them as indistinguishable
         * during the main algorithm.
         */
        auto eq_context = [&](size_t i, size_t j) {
            if (item_context[i] != item_context[j]) {
                return false;
            }
            for (size_t ci = 0; ci < context_to_string.size(); ++ci) {
                if (join_overlap[ci][i] != join_overlap[ci][j]) {
                    return false;
                }
            }
            return true;
        };
        vector <size_t> eq_context_group(N_search, size_t(-1));
        for (size_t si = 0; si < N_search; ++si) {
            for (size_t sj = si-1; sj+1 > 0; --sj) {
                size_t i = search_items[si], j = search_items[sj];
                if (!contextfree[j] && eq_context(i, j)) {
                    DEBUG("  eq context: %s =c= %s\n", items[i].c_str(), items[j].c_str());
                    eq_context_group[si] = sj;
                    break;
                }
            }
        }

        // Figure out the combined context size. A combined context has
        // one entry for each context-free item plus one for context0.
        size_t ccontext_size = N - N_search + 1;

        // Assert that various parameters all fit into our data types
        using ccontext_t = bitset_t;
        assert (context_to_string.size() + ccontext_size <= bitset_bits);
        assert (N_search <= solver_max_item_set);
        assert (base_length < 0xff);

        // Initial combined context.
        unordered_map <size_t, size_t> cc0_full;
        ++cc0_full[context0_index];
        for (size_t i = 0; i < N; ++i) {
            if (contextfree[i]) {
                ++cc0_full[item_context[i]];
            }
        }
        // Now pack into unary-encoded bitset. The bitset stores the
        // count for each context as <count> number of 0 bits,
        // followed by a 1 bit.
        ccontext_t cc0 = 0;
        for (size_t ci = 0, b = 0; ci < context_to_string.size(); ++ci, ++b) {
            b += cc0_full[ci];
            cc0 |= ccontext_t(1) << b;
        }

        // Map from (item set, context) to maximum achievable overlap
        Solver_table k_solns;
        // Base case: cc0 with empty set
        k_solns.update_score(0, cc0, 0);

        // Now start dynamic programming. k is current subset size
        size_t eq_context_groups = 0;
        for (size_t g: eq_context_group) eq_context_groups += (g != size_t(-1));
        if (context0.empty()) {
            INFO("solve: N=%zu, N_search=%zu, ccontext_size=%zu, #contexts=%zu, #eq_context_groups=%zu\n",
                 N, N_search, ccontext_size, context_to_string.size(), eq_context_groups);
        } else {
            DEBUG("solve: context=%s, N=%zu, N_search=%zu, ccontext_size=%zu, #contexts=%zu, #eq_context_groups=%zu\n",
                  context0.c_str(), N, N_search, ccontext_size, context_to_string.size(), eq_context_groups);
        }
        for (size_t k = 0; k < N_search; ++k) {
            decltype(k_solns) k1_solns;

            // The main bottleneck of this program is updating k1_solns,
            // which (for larger N) becomes a huge table.
            // We use a prefetch queue to reduce memory latency.
            const size_t prefetch = 8;
            array <Solver_entry, prefetch> entry_queue;
            size_t update_i = 0;

            // Iterate every k-subset
            for (Solver_entry entry: k_solns.t) {
                if (entry.is_empty()) continue;

                bitset_t s = entry.items;
                ccontext_t ccontext = entry.context;
                size_t overlap = entry.score;

                // Try adding a new item
                for (size_t si = 0; si < N_search; ++si) {
                    bitset_t s1 = s | bitset_t(1) << si;
                    if (s == s1) {
                        continue;
                    }
                    // Add items in each eq_context_group sequentially
                    if (eq_context_group[si] != size_t(-1) &&
                        !(s & bitset_t(1) << eq_context_group[si])) {
                        continue;
                    }
                    size_t i = search_items[si]; // actual item index

                    size_t new_context = item_context[i];
                    // Increment ccontext's count for new_context.
                    // We need to find its delimiter 1 bit
                    size_t bit_n = bit_select(ccontext, new_context);
                    ccontext_t ccontext_n =
                        ((ccontext & ((ccontext_t(1) << bit_n) - 1))
                         | ((ccontext >> bit_n << (bit_n + 1))));

                    // Select non-empty sub-contexts to substitute for new_context
                    for (size_t ci = 0, bit1 = 0, count;
                         ci < context_to_string.size();
                         ++ci, bit1 += count + 1)
                    {
                        assert (ccontext_n >> bit1);
                        count = __builtin_ctzll(ccontext_n >> bit1);
                        if (!count
                            // We just added new_context; we can only remove an existing
                            // context entry there i.e. there must be at least two now
                            || (ci == new_context && count < 2)) {
                            continue;
                        }

                        // Decrement ci in ccontext_n
                        bitset_t ccontext1 =
                            ((ccontext_n & ((ccontext_t(1) << bit1) - 1))
                             | ((ccontext_n >> (bit1 + 1)) << bit1));

                        size_t overlap1 = overlap + join_overlap[ci][i];

                        // do previous prefetched update
                        if (update_i >= prefetch) {
                            Solver_entry entry = entry_queue[update_i % prefetch];
                            k1_solns.update_score(entry.items, entry.context, entry.score);
                        }

                        // queue the current update and prefetch
                        Solver_entry entry1;
                        size_t probe_index = k1_solns.probe_index(Solver_table::entry_hash(s1, ccontext1));
                        __builtin_prefetch(&k1_solns.t[probe_index]);
                        entry1.items = s1;
                        entry1.context = ccontext1;
                        entry1.score = overlap1;
                        entry_queue[update_i % prefetch] = entry1;

                        ++update_i;
                    }
                }
            }

            // do remaining queued updates
            for (size_t j = 0; j < min(update_i, prefetch); ++j) {
                Solver_entry entry = entry_queue[j];
                k1_solns.update_score(entry.items, entry.context, entry.score);
            }

            if (context0.empty()) {
                INFO("  hash stats: |solns[%zu]| = %zu, %zu lookups, %zu probes\n",
                     k+1, k1_solns.size(), update_i, k1_solns.num_probes_);
            } else {
                DEBUG("  hash stats: |solns[%zu]| = %zu, %zu lookups, %zu probes\n",
                      k+1, k1_solns.size(), update_i, k1_solns.num_probes_);
            }
            k_solns.swap(k1_solns);
        }

        // Overall solution
        size_t max_overlap = 0;
        for (Solver_entry entry: k_solns.t) {
            if (entry.is_empty()) continue;
            max_overlap = max(max_overlap, size_t(entry.score));
        }
        return base_length - max_overlap;
    }
};

// Wrapper for Solver that also finds the smallest solution string
string smallest_containing_string(vector <string> items)
{
    items = dedup_items("", items);

    size_t soln_length;
    {
        Solver solver(items, "");
        soln_length = solver.solve();
    }
    DEBUG("Found solution length: %zu\n", soln_length);

    string soln;
    vector <string> remaining_items = items;
    while (remaining_items.size() > 1) {
        // Add all possible next items, in lexicographic order
        vector <pair <string, size_t>> next_solns;
        for (size_t i = 0; i < remaining_items.size(); ++i) {
            const string& item = remaining_items[i];
            next_solns.push_back(make_pair(join_strings(soln, item), i));
        }
        assert (next_solns.size() == remaining_items.size());
        sort(next_solns.begin(), next_solns.end());

        // Now try every item in order
        bool found_next = false;
        for (auto ns: next_solns) {
            size_t i;
            string next_soln;
            tie(next_soln, i) = ns;
            DEBUG("Trying: %s + %s -> %s\n",
                  soln.c_str(), remaining_items[i].c_str(), next_soln.c_str());
            vector <string> next_remaining;
            for (size_t j = 0; j < remaining_items.size(); ++j) {
                if (next_soln.find(remaining_items[j]) == string::npos) {
                    next_remaining.push_back(remaining_items[j]);
                }
            }

            Solver solver(next_remaining, next_soln);
            size_t next_size = solver.solve();
            DEBUG("  ... next_size: %zu + %zu =?= %zu\n", next_size, next_soln.size(), soln_length);
            if (next_size + next_soln.size() == soln_length) {
                INFO("  found next item: %s\n", remaining_items[i].c_str());
                soln = next_soln;
                remaining_items = next_remaining;
                // found lexicographically smallest solution, break now
                found_next = true;
                break;
            }
        }
        assert (found_next);
    }
    soln = join_strings(soln, remaining_items[0]);

    return soln;
}

int main()
{
    string prev_soln;
    vector <string> items;
    size_t p = 1;
    for (size_t N = 1;; ++N) {
        for (++p; items.size() < N; ++p) {
            if (is_prime(p)) {
                char buf[99];
                snprintf(buf, sizeof buf, "%zu", p);
                items.push_back(buf);
                break;
            }
        }

        // Try to reuse previous solution (this works for N=11,30,32...)
        string soln;
        if (prev_soln.find(items.back()) != string::npos) {
            soln = prev_soln;
        } else {
            soln = smallest_containing_string(items);
        }
        printf("%s\n", soln.c_str());
        prev_soln = soln;
    }
}

在线尝试！

以及TIO上的Prime专用版本。抱歉，但是我没有参加这些课程，而且帖子时长有限制。

JavaScript（Node.js），在241秒内获得24分

结果

• $a(1)$$a(21)$
• $a(22)=231129413434717353759619679$
• $a(23)=23112941343471735359619678379$
• $a(1)$$a(24)$

算法

这是一种递归搜索，它尝试将数字合并在一起的所有可能方式，并在到达叶节点时最终按照字典顺序对结果列表进行排序。

$x$$y$$k$$x$$k$$y$$k$$y$$k$$x$

在每次迭代的开始，将从列表中删除在另一个条目中可以找到的任何条目。

通过跟踪访问的节点可以显着提高速度，因此当不同的操作导致相同的列表时，我们可以提早中止操作。

通过尽可能地更新和还原列表，而不是生成一个匿名用户 Neil 所建议的副本，可以稍微提高速度。

例

$n=7$$[2,3,5,7,11,13,17]$

[]                        // start with an empty list
[ 2 ]                     // append 2
[ 2, 3 ]                  // append 3
[ 2, 3, 5 ]               // append 5
[ 2, 3, 5, 7 ]            // append 7
[ 2, 3, 5, 7, 11 ]        // append 11
[ 2, 3, 5, 7, 11, 13 ]    // append 13
[ 2, 5, 7, 11, 13 ]       // remove 3, which appears in 13
  [ 2, 5, 7, 113, 13 ]    //   try to merge 11 and 13 into 113
  [ 2, 5, 7, 113 ]        //   remove 13, which now appears in 113
  [ 2, 5, 7, 113, 17 ]    //   append 17
  [ 2, 5, 113, 17 ]       //   remove 7, which appears in 17
  --> leaf node: 1131725  //   new best result
[ 2, 5, 7, 11, 13, 17 ]   // append 17
[ 2, 5, 11, 13, 17 ]      // remove 7, which appears in 17
  [ 2, 5, 113, 13, 17 ]   //   try to merge 11 and 13 into 113
  [ 2, 5, 113, 17 ]       //   remove 13, which now appears in 113
                          //   abort because this node was already visited
                          //   (it was a leaf node anyway, so we don't save much here)
  [ 2, 5, 117, 13, 17 ]   //   try to merge 11 and 17 into 117
  [ 2, 5, 117, 13 ]       //   remove 17, which now appears in 117
  --> leaf node: 1171325  //   not better than the previous one
--> leaf node: 11131725   // not better than the previous one

码

在线尝试！

let f = n => {
  let visited = {},
      a, d, k, best, search;

  // build the list of primes, as strings
  for(a = [ '2' ], n--, k = 3; n; k++) {
    for(d = k; k % (d -= 2);) {}
    d == 1 && n-- && a.push(k + '');
  }

  best = a.join('');

  // recursive search function
  (search = (a, n = 0, r = []) => {
    let x, y, i, j, k, s;

    // remove all entries in r[] that can be found in another entry
    r = r.filter((p, i) => !r.some((q, j) => i != j && ~q.indexOf(p)));

    // abort early if this node was already visited
    if(visited[r]) {
      return;
    }

    // otherwise, mark it as visited
    visited[r] = 1;

    // walk through all distinct pairs (x, y) in r[]
    for(i = 0; i < r.length; i++) {
      for(j = i + 1; j < r.length; j++) {
        x = r[i];
        y = r[j];

        // try to merge x and y if:
        // 1) the first k digits of x equal the last k digits of y
        for(k = 1; x.slice(0, k) == y.slice(-k); k++) {
          r[i] = y + x.slice(k);
          search(a, n, r);
        }

        // or:
        // 2) the first k digits of y equal the last k digits of x
        for(k = 1; y.slice(0, k) == x.slice(-k); k++) {
          r[i] = x + y.slice(k);
          search(a, n, r);
        }
        r[i] = x;
      }
    }

    if(x = a[n]) {
      // there are other primes to process, so go on with the next one
      search(a, n + 1, [...r, x]);
    }
    else {
      // this is a leaf node: see if we've improved our current score
      s = r.join('');

      if(s.length <= best.length) {
        s = r.sort().join('');

        if(s.length < best.length || s < best) {
          best = s;
        }
      }
    }
  })(a);

  return best;
}

协和TSP解算器，299秒得分84

好吧……我只是现在才意识到这一点而感到傻。

这整件事本质上是一个旅行推销员的问题。对于每对素数p和q，添加一条边线，其权重为q（加上重叠的数字）相加的位数。此外，为每个素数添加一个初始边p，其权重为的长度p。最短的行销员路径与最小的主要收容编号的长度匹配。

然后，诸如Concorde之类的工业级TSP求解器将使该问题的工作时间缩短。

此项可能被认为是非竞争性的。

结果

求解器N=350大约需要20个CPU小时。完整的结果对于一个SE职位来说太长了，而OEIS无论如何都不需要那么多的术语。这是前200个：

1 2
2 23
3 235
4 2357
5 112357
6 113257
7 1131725
8 113171925
9 1131719235
10 113171923295
11 113171923295
12 1131719237295
13 11317237294195
14 1131723294194375
15 113172329419437475
16 1131723294194347537
17 113172329419434753759
18 2311329417434753759619
19 231132941743475375961967
20 2311294134347175375961967
21 23112941343471735375961967
22 231129413434717353759619679
23 23112941343471735359619678379
24 2311294134347173535961967837989
25 23112941343471735359619678378979
26 2310112941343471735359619678378979
27 231010329411343471735359619678378979
28 101031071132329417343475359619678378979
29 101031071091132329417343475359619678378979
30 101031071091132329417343475359619678378979
31 101031071091131272329417343475359619678378979
32 101031071091131272329417343475359619678378979
33 10103107109113127137232941734347535961967838979
34 10103107109113127137139232941734347535961967838979
35 10103107109113127137139149232941734347535961967838979
36 1010310710911312713713914923294151734347535961967838979
37 1010310710911312713713914915157232941734347535961967838979
38 1010310710911312713713914915157163232941734347535961967838979
39 10103107109113127137139149151571631672329417343475359619798389
40 10103107109113127137139149151571631672329417343475359619798389
41 1010310710911312713713914915157163167173232941794347535961978389
42 101031071091131271371391491515716316717323294179434753596181978389
43 101031071091131271371391491515716316723294173434753596181917978389
44 101031071091131271371391491515716316717323294179434753596181919383897
45 10103107109113127137139149151571631671731792329418191934347535961978389
46 10103107109113127137139149151571631671731791819193232941974347535961998389
47 101031071091271313714915157163167173179181919321139232941974347535961998389
48 1010310710912713137149151571631671731791819193211392232941974347535961998389
49 1010310710912713137149151571631671731791819193211392232272941974347535961998389
50 10103107109127131371491515716316717317918191932113922322722941974347535961998389
51 101031071091271313714915157163167173179181919321139223322722941974347535961998389
52 101031071091271313714915157163167173179181919321139223322722923941974347535961998389
53 1010310710912713137149151571631671731791819193211392233227229239241974347535961998389
54 101031071091271313714915157163167173179211392233227229239241819193251974347535961998389
55 101031071091271313714915157163167173179211392233227229239241819193251972574347535961998389
56 101031071091271313714915157163167173179211392233227229239241819193251972572634347535961998389
57 101031071091271313714915157163167173179211392233227229239241819193251972572632694347535961998389
58 101031071091271313714915157163167173179211392233227229239241819193251972572632694347535961998389
59 1010310710912713137149151571631671731792113922332277229239241819193251972572632694347535961998389
60 101031071091271313714915157163167173211392233227722923924179251819193257263269281974347535961998389
61 1010310710912713137149151571631671732113922332277229239241792518191932572632692819728343475359619989
62 10103107109127131371491515716316717321139223322772293239241792518191932572632692819728343475359619989
63 1010307107109127131371491515716316717321139223322772293239241792518191932572632692819728343475359619989
64 10103071071091271311371391491515716316721173223322772293239241792518191932572632692819728343475359619989
65 10103071071091271311371491515716313916721173223322772293239241792518191932572632692819728343475359619989
66 10103071071091271311371491515716313921167223317322772293239241792518191932572632692819728343475359619989
67 10103071071091271311371491515716313921167223317322772293239241792518191932572632692819728343475359619989
68 1010307107109127131137149151571631392116722331732277229323924179251819193257263269281972833743475359619989
69 1010307107109127131137149151571631392116722331732277229323924179251819193257263269281972833743475359619989
70 101030710710912713113714915157163139211672233173227722932392417925181919325726326928197283374347534959619989
71 101030710710912713113714915157163139211672233173227722932392417925181919325726337269281972834743534959619989
72 101030710710912713113714915157163139211672233173227722932392417925181919337257263472692819728349435359619989
73 10103071071091271311371491515716313921167223317322772293372392417925181919347257263492692819728353594367619989
74 101030710710912713113714915157163139211672233173227722932392417925181919337347257263492692819728353594367619989
75 1010307107109127131137313914915157163211672233173227722933792392417925181919347257263492692819728353594367619989
76 101030710710912713113731391491515716321167223317322772293379239241792518191934725726349269281972835359438367619989
77 101030710710912713113731391491515716321167223317337922772293472392417925181919349257263535926928197283674383896199
78 1010307107109127131137313914915157163211672233173379227722934723972417925181919349257263535926928197283674383896199
79 101030710710912713113731391491515721163223317337922772293472397241672517925726349269281819193535928367401974383896199
80 101030710710912713113731391491515721163223317337922772293472397241672517925726349269281819193535928367401974094383896199
81 101030710710912713113731391491515721163223317337922772293472397241916725179257263492692818193535928367401974094383896199
82 1010307107109127131137313914915157223317322772293379239724191634725167257263492692817928353594018193674094211974383896199
83 1010307107109127131137313914922331515722772293379239724191634725167257263492692817353592836740181938389409421197431796199
84 101030710710912713113731391492233151572277229323972419163472516725726349269281735359283674018193838940942119743179433796199
85 101030710710912713113731391492233151572277229323924191634725167257263492692817353592836740181938389409421197431794337943976199
86 1010307107109127131137313914922331515722772293239241916347251672572634926928173535928367401819383894094211974317943379443976199
87 1010307107109127131137313914922331515722772293239241916347251672572634926928173535928367401819383894094211974317943379443974496199
88 1010307107109127131137313914922331515722772293239241916347251672572634926928173535928367401819383894094211974317943379443974494576199
89 10103071071091271311373139149223315157227722932392419163472516725726349269281735359283674018193838940942119743179433794439744945746199
90 10103071071091271311373139149223315157227722932392419163251672572634726928173492835359401819367409421197431794337944397449457461994638389
91 10103071071091271311373139149223315157227722932392419163251672572634726928173492835359401819367409421197431794337944397449457461994638389467
92 101030710710912713113731391492233151572277229323924191632516725726347926928173492835359401819367409421197431794337944397449457461994638389467
93 101030710710912713113731391492233151572277229323924191632516725726347926928173492835359401819367409421197431794337944397449457461994638389467487
94 101030710710912713113731392233149151572277229323924191632516725726347926928173492835359401819367409421197431794337944397449457461994638389467487
95 1010307107109127131137313922331491515722772293239241916325167257263479269281734928353594018193674094211974317943379443974499457461994638389467487
96 1010307107109127131137313922331491515722772293239241916325167257263269281734792834940181935359409421197431794337944397449945746199463674674875038389
97 1010307107109127131137313922331491515722772293239241916325167257263269281734792834940181935359409421197431794337944397449945746199463674674875038389509
98 101030710710912713113732233139227722932392419149151572516325726326928167283479401734940942118193535943179433794439744994574619746367467487503838950952199
99 1010307107109127131137322331392277229324191491515725163257263269281672834794017349409421181935359431794337944394499457461974636746748750383895095219952397
100 101030710710922331127131373227722932414915157251632572632692816728347940173494094211394317943379443944994574618191935359463674674875038389509521975239754199
101 101030710710922331127131373227722932414915157251632572632692816728347401734940942113943179433794439449945746181919353594636746748750383895095219752397541995479
102 101030710710922331127131373227722932414915157251632572632692816728347401734940942113943179433794439449945746181919353594636746748750383895095219752397541995479557
103 101030710710922331127131373227722932414915157251632572632692816728340173474094211394317943379443944945746181919349946353594674875036750952197523975419954795575638389
104 101030710710922331127131373227722932414915157251632572632692816728340173474094211394317943379443944945746181919349946353594674875036750952197523975419954795575638389569
105 101030710722331109227127722932413137325149151571632572632692816728340173474094211394317943379443944945746181919349946353594674875036750952197523975419954795575638389569
106 1010307107223311092271277229324131373251491515716325726326928167283401734740942113943179433794439449457461819193499463535946748750367509521975239754199547955775638389569
107 1010307107223311092271277229324131373251491515716325726326928167283401734740942113943179433794439449457461819193499463535946748750367509521975239754199547955775638389569587
108 10103071072233110922712772293241313732514915157163257263269281672834017340942113943179433794439449457461819193474634994674875035359367509521975239754199547955775638389569587
109 10103071072233110922712772293241313732514915157163257263269281672834017340942113943179433794439449457461819193474634994674875035359367509521975239754199547955775638389569587599
110 1010307223311072271092293241277251313732571491515726326928163283401674094211394317343379443944945746179463474674875034995095218191935359367523975419754795577563838956958759960199
111 1010307223311072271092293241277251313732571491515726326928163283401674094211394317343379443944945746179463474674875034995095218191935359367523975419754795577563838956958759960199607
112 1010307223311072271092293241277251491515716325726326928167283401734094211313734317943379443944945746139463474674875034995095218191935359367523975419754795577563838956958759960199607
113 22331101030722710722932410925127725714915157263269281632834016740942113137343173433794439449457461394634746748750349950952181919353593675239754197547955775638389569587599601996076179
114 2233110103072271072293241092512571277263269281491515728340163409421131373431734337944394494574613946347467487503499509521675239754191819353593675479557756383895695875996019760761796199
115 22331010307227107229324109251257126311277269281491515728340163409421131373431734337944394494574613946347467487503499509521675239754191819353593675479557756383895695875996019760761796199
116 22331010307227107229324109251257126311269281277283401491515740942113137343173433794439449457461394634674875034750952163499523975416754795577563535936756958759960181919383896076179619764199
117 223310103072271072293241092512571263112692812772834014915157409421131373431734433794494574613946346748750347509521634995239541675479557756353593675695875996018191938389607617961976419964397
118 223310103072271072293241092512571263112692812772834014915157409421131373431734433794494574613946346748750347509521634995239541675475577563535936756958759960181919383896076179619764199643976479
119 223310103072271072293241092512571263112692812772834014915157409421131373431734433794494574613946346748750347509521634995239541675475577563535695875935996018191936760761796197641996439764796538389
120 2233101030722710722932410925125712631126928127728340149151574094211313734317344337944945746139463467487503475095216349952395416754755775635356958760181919359367607617961976419964397647965383896599
121 22331010307227107229324109251257126311269281277283401491515740942113137343173443379449457461394634674875034750952163499523954167547557756353569587601819193593676076179641976439764796538389659966199
122 223310103072271072293241092512571263112692812772834014915157409421131373431734433794494574613946346734748750349950952163523954167547557756353569587601819193593676076179641976439764796538389659966199
123 2233101030722710722932410925125712631126928127728340149151574094211313734317344337944945746139463467347487503499509521635239541675475577563535695876018191935936776076179641976439764796538389659966199
124 2233101030722710722932410925125712631126928127728340149151574094211313734317344337944945746139463467347487503499509521635239541675475577563535695876018191935936076179641976439764796536776599661996838389
125 22331010307227107229324109251257126311269127728128340149151574094211313734317344337944945746139463467347487503499509521635239541675475577563535695876018191935936076179641976439764796536776599661996838389
126 2233101030701072271092293241251257126311269127728128340149151574094211313734317344337944945746139463467347487503499509521635239541675475577563535695876018191935936076179641976439764796536776599661996838389
127 223310103070107092271092293241251257126311269127728128340149151574094211313734317344337944945746139463467347487503499509521635239541675475577563535695876018191935936076179641976439764796536776599661996838389
128 223310103070107092271092293241251257191263112691277281283401491515740942113137343173443379449457461394634673474875034995095216352395416754755775635356958760181935936076179641976439764796536776599661996838389
129 22331010307010709227109229324125125719126311269127277281283401491515740942113137343173443379449457461394634673474875034995095216352395416754755775635356958760181935936076179641976439764796536776599661996838389
130 223307010103227092293241072510925712631126912719128128340140942113137331491515727743173443379449457461394634673474875034995095216352395416754755775635356958760181935936076179641976439764796536776599661996838389
131 2233070101032270922932410725109257126311269127191281283401409421131373314915157277431734433794494574613946346739487503475095216349952395416754755775635356958760181935936076179641976439764796536776599661996838389
132 2233070101032270922932410725109257126311269127191281283401409421131373314915157277431734433794494574613946346739487503475095216349952395416754755775635356958760181935936076179641976439764796536776599661996838389
133 223307010103227092293241072510925712631126912719128128340140942113137331443173449149457277433794613946346739487503475095215157516349952395416754755775635356958760181935936076179641976439764796536776599661996838389
134 22330701010322709229324107251092571263112691271912812834014094211313733144317344914945727743379461394634673948750347509521515751634995239541675475575635356958757760181935936076179641976439764796536776599661996838389
135 22330701010322709229324107251092571263112691271912812834014094211313733144317344914945727743379461394634673948750347509521515751634995239541675475575635356958757760181935936076179641976439764796536776599661996838389
136 2233070101032270922932410725109257126311269127191281283401409421131373314431734491494572774337946139463467394875034750952151575163499523954167547557563535695875776018193593607617964197643976479653677696599661996838389
137 22330701010322709229324107251092571263112691271912812834014094211313733144317344914945727734613946346739487433795034750952151575163499523954167547557563535695875776018193593607617964197643976479653677696599661996838389
138 2233070101032270922932410725109257126311269127191281283401409421131373314431734491494572773461394634673948743379503475095215157516349952395416754755756353569587577601819359360761796419764397647965367787696599661996838389
139 22330701010322709229324107251092571263112691271912812834014094211313733144317344914945727734613946346739487433795034750952151575163499523954167547557563535695875776018193593607617964197643976479765367787696599661996838389
140 22330701010322709229324107251092571263112691271912812834014094211313733144317344914945727734613946346739487433795034750952151575163499523954167547557563535695875776018193593607617964197643976479765367787696599661996838389809
141 223307010103227092293241072510925712631126912719128112834014094211313733144317344914945727734613946346739487433795034750952151575163499523954167547557563535695875776018193593607617964197643976479765367787696599661996838389809
142 223307010103227092293241072510925712631126912719128112834014094211313733144317344914572773461394634673948743379503475095214952395415157516349954755756353569587577601676076179641935936439764797653677659966197876968383898098218199
143 223070101032270922932410725109257126311269127191281128340140942113137331443173449145727734613946346739487433475034950952149952337954151575163535475575635695875776016760761796419359364396479765367765996619768383898098218199823978769
144 223070101032270922932410725109257126311269127191281128340140942113137331443173449145727433461394634673474875034950952149952337954151575163535475575635695875773960167607617964193593643964797653677659966197683838980982181998239769827787
145 223070101032270922924107251092571263112691271912811283401409421131373314431734491457274334613946346734748750349509521499523379541515751635354755756356958757739601676076179641935936439647976536599661976836776980982181998239782778782938389
146 2230701010322709229241072510925712631126912719128112834014094211313733144317344914572743346139463467347487503495095214995233795415157516353547557563569587577396016760761796419359364396479765367765996619768383976980982181998239827787829389
147 2230701010322709229241072510925712631126912719128112834014094211313733144317344914572743346139463467347487503495095214995233795415157516353547557563569587577396016760761796419359364396479765365996619768367769809821819982397827787829383985389
148 2230701010322709229241072510925712631126912719128112834014094211313733144317344914572743346139463467347487503495095214995233795415157516353547557563569587576016760761796419359364396479765365996619768367739769809821819982398277829383985389857787
149 2230701010322709229241072510925712631126912719128112834014094211313733144317344914572743346139463467347487503495095214995233795415157516353547557563569587576016760761796419359364396479765365966197683677397698098218199823982778293839853898577878599
150 2230701010322709229241072510925712631126912719128112834014094211313733144317344914572743346139463467347487503495095214995233795415157516353547557563569587576016760761796419359364396479765365966197683677397698098218199823982778293839853857787859986389
151 22307010103227092292410725109257126311269127191281128340140942113137331443173449145727433461394634673474875034950952149952337954151575163535475575635695875760167607617964193593643964797653659661976836773976980982181998239827782938398538577877859986389
152 22307010103227092292410725109257126311269127191281128340140942113137331443173449145727433461394634673474875034950952149952337954151547515755756353569587576016359360761796419364396479765365966197683676980982167739782398277829383985385778778599863898818199
153 22307010103227092292410725109257126311269127191281128340140942113137331443173449145727433461394634673474875034950952149952337954151547515755756353569587576016359360761796419364396479765365966197683676980982167739782398277829383853857787785998638988181998839
154 22307010103227092292410725109257126311269127191281128340140942113137331443173449145727433461394634673474875034950952149952337954151547515755756353569587576016359360761796419364396479765365966197683676980982167739782398277829383853857785998638988181998839887787
155 2230701010322709072292410725109257126311269127191281128340140942113137331443173449145727433461394634673474875034950952149952337954151547515755756353569587576016359360761796419364396479765365966197683676980982167739782398277829383853857785998638988181998839887787
156 22307010103227090722924107251092571263112691127191281128340140942113137331443173449145727433461394634673474875034950952149952337954151547515755756353569587576016359360761796419364396479765365966197683676980982167739782398277829383853857785998638988181998839887787
157 22307010103227090722924107251092571263112691127191281128340140942113137331443173449193457274334613946346734748750349509521499523379541515475155756353569587576015760761796419764396479765359365966199683676980982163823978277398293838538577859986389881816778778839887
158 2230701010322709072292410725109257126311269112719128112834014092934211313733144317344919345727433461394634673474875034950952149952337954151547515575635356958757601576076179641976439647976535936596619968367698098216382397827739829853838577859986389881816778778839887
159 22307010103227090722924107251092571263112691127191281128340140929342113137274314433173344919345746139463467347487503495095214995233735354151547515575635695875760157607617964197643964796535937976596619968367698098216382397827739829853838577859986389881816778778839887
160 2230701010322709072292410725109257126311269112719128112834014092934211313727431443317334491934574613941463467347487503495095214995233735354151547515575635695875760157607617964197643964796535937976596619968367698098216382397827739829853838577859986389881816778778839887
161 223070101032270907229241072510925712631126911271912811283401409293421131372743144331733449193457461394146346734748750349475095214995233735354151547515575635695875760157607617964197643964796535937976596619968367698098216382397827739829853838577859986389881816778778839887
162 22307010103227090722924107251092571263112691127191281128340140929342113137274314433173344919345746139414634673474875034947509521499523373535415154751557563569535875760157607617964197643964796535937976596619968367698098216382397827739829853838577859986389881816778778839887
163 2230701010322709072292410725109257126311269112719128112834014092934211313727431443317334491934574613941463467347487503494750952149952337353541515475155756356953587576015760761796419764396479653593797659661996768367698098216382397827739829853838577859986389881816778778839887
164 22307010103227090722924107251092571263112691127128112834014092934211313727431443317334491457461394146346734748750349475095214995233735354151547515575635695358757601576076179641919359379643964797197653659661996768367698098216382397827739829853838577859986389881816778778839887
165 223070101032270907229241072510925712631126911271281128340140929342113137274314433173344914574613941463467347487503494750952149952337353541515475155756356953587576015760761796419193593796439647971976536596619967683676980982163823977398277829853838577859986389881816778778839887
166 22307010103227090722924107251092571263112691127128112834014092934211313727431443317334491457461394146346734748750349475095214995233735354151547515575635695358757601576076179641919359379643964797197653659661996768367698098216382397739827782983838538577859986389881816778778839887
167 223070101032270907229241072510925712631126911271281128340140929342113137274314433173344914574613941463467347487503494750952149915152337353541547515575635695358757601576076179641919359379643964797197653659661996768367698098216382397739827782983838538577859986389881816778778839887
168 2230701010322709072292410725109257126311269112712811283401409293421131372743144331733449145746139414634673474875034947509521499151523373535415475155756356953587576015760761796419193593796439647971976536596619967683676980982163823977398277829838385385778599786389881816778778839887
169 2230701009070922710103229241072510925712631126911272728112834014092934211313733144317344914574334613941463467347487503494750952149915152337515415475575635356953587576015760761796419193593796439647971976536596619967683676980982163823977398277829838385385778599786389881816778778839887
170 22307010090709227101310322924107251092571263112691127272811283401409293421134431373317344914574334613941463467347487503494750952149915152337515415475575635356953587576015760761796419193593796439647971976536596619967683676980982163823977398277829838385385778599786389881816778778839887
171 22307010090709227101310191032292410725109257126311269112727281128340140929342113443137331734491457433461394146346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
172 22307010090709227101310191021032292410725109257126311269112727281128340140929342113443137331734491457433461394146346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
173 223070100907092271013101910210310722924109251257126311269112727281128340140929342113443137331734491457433461394146346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
174 223070100907092271013101910210310331107229241092512571263132691127272811283401409293421137334431734491457433461394146346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
175 223070100907092271013101910210310331103922924107251092571263132691127272811283401409293421137334431734491457433461394146346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
176 223070100907092271013101910210310331103922924104910725109257126313269112727281128340140929342113733443173449414574334613946346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
177 223070100907092271013101910210310331103922924104910510725109257126313269112727281128340140929342113733443173449414574334613946346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
178 223070100907092271013101910210310331103922924104910510610725109257126313269112727281128340140929342113733443173449414574334613946346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
179 223070100907092271013101910210310331103922924104910510610631325107257109263269112727281128340140929342113733443173449414574334613946346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
180 223070100907092271013101910210310331103922924104910510610631325106911072571092632692811272728340140929342113733443173449414574334613946346734748750349475095214991935233751515415475575635356953587576015760761796419643964796535937971976596619967683676980982163823977398277829838385385778599786389881816778778839887
181 223070100907092271013101910210310331103922924104910510610631325106911072571087263269281092834012727409293421137334431734494145743346139463467347487503494750952149919352337515154154755756353569535875760157607617964196439647965359379719765966199676836769809821638239773982778298383853857785997863898811816778778839887
182 2230701009070922710131019102103103311039229241049105106106313251069107257108726326928109112727283401409293421137334431734494145743346139463467347487503494750952149919352337515154154755756353569535875760157607617964196439647965359379719765966199676836769809821638239773982778298383853857785997863898811816778778839887
183 2230701009070922710131019102103103311039229241049105106106313251069107257108726326928109110932834012727409293421137334431734494145743346139463467347487503494750952149919352337515154154755756353569535875760157607617964196439647965359379719765966199676836769809821638239773982778298383853857785997863898811816778778839887
184 2230701009070922710131019102103103311039229241049105106106313251069107257108726326928109110932834010971929340941272742113733443173449457433461394634673474875034947509521499193523375151541547557563535695358757601576076179641976439647965359379765966199676836769809821638239773982778298383853857785997863898811816778778839887
185 2230701009070922710131019102103103311039229241049105106106313251069107257108726326928109110932834010971929340941272742113733443173449457433461394634673474875034947509521499193523375151541547557563535695358757601576076179641976439647965359379765966199676836769809821638239773982778298383853857785997863898811816778778839887
186 2230701009070922710131019102103103311039229241049105106106313251069107257108726326928109110932834010971929340941272742113733443173449457433461394634673474875034947509521499193523375151541547557563535695358757601576076179641976439647965359379765966199676836769809821638239773982778298383853857785997863898811816778778839887
187 223070100907092271013101910210310331103922924104910510610631325106910725710872632692810911093283401097192934094127274211173344317433449457461373463467347487503494750952149919352337515154157547557563535695358757601635937960761796419764396479765365966199676836769809821677397782398277829838385385778599786389881811398839887787
188 223070100907092271013101910210310331103922924104910510610631325106910725710872632692810911093283401097192934094111727421123344317334494574337346137463467347487503494750952127514991935235354151575475575635695358757601635937960761796419764396479765365966199676836769809821677397782398277829838385385778599786389881811398839887787
189 1009070101307092232271019102103103310491051061063110392292410691072510872571091109326326928109719283401117274092934211233443131733449411294574337346137463467347487503494750952127514991935235354151575475575635695358757601635937960761796419764396479765365966199676836769809821677397782398277829838385385778599786389881811398839887787
190 10090701013070922322710191021031033104910510610631103922924106910725108725710911093263269281097192834011172740929342112334431317334494112945743373461374634673474875034947509521139523535412751499193547557563569535875760157607617964197643964796535937976596619967683676980982163823977398277829838385385778599786389881151816778778839887
191 100907010130709101910210310331049105106106311039223227106910722924108725109110932571097192632692811172728340112334092934211294113137334431734494574337461394634673474875034947509521151153523535412751499193547557563569535875760157607617964197643964796535937976596619967683676980982163823977398277829838385385778599786389881816778778839887
192 1009070101307091019102103103310491051061063110392232271069107229241087251091109325710971926326928111727283401123340929342112941131373344317344945743374613946346734748750349475095211511535235354116354751275575635695358757601499193593796076179641976439647976536596619967683676980982157739778239827782983838538578599786389881816778778839887
193 1009070101307092232271019102103103310491051061063110392292410691072510872571091109326326928109711171928340112334092934211294113137274317334433734494574613946346734748750349475095211511535235354127514991935475575635695358757601576076179641976439647965359379765966199676836769809821677397782398277829838385385778599786388181163898839887787
194 10090701013070922322710191021031033104910510610631103922924106910725108725710911093263269281097111719283401123340929342112941131372743173344337344945746139463467347487503494750952115115352353541163547512755756356953587576014991935937960761796419764396479765365966199676836769809821577397782398277829838385385785997863898811816778778839887
195 100907010130709101910210310331049105106106311039223227106910722924108725109110932571097111719263269281123283401129293409411313727421151153443173344945743346139463467347487503494750952116352337353541181187512754755756356953587576014991935937960761796419764396479765365966199676836769809821577397782398277829838385385785997863898816778778839887
196 100907010130709101910210310331049105106106310691072231103922710872292410911093251097111711232571926326928112928340113137274092934211511534431733449411634574334613946346734748750349475095211811875119352337353541275475575635695358757601499196076179641976439647965359379765966199676836769809821577397782398277829838385385785997863898816778778839887
197 100907010130709101910210310331049105106106310691072231103922710872292410911093251097111711232571926326928112928340113137274092934211511534431733449411634574334613946346734748750349475095211811875119352337353541201275475575635695358757601499196076179641976439647965359379765966199676836769809821577397782398277829838385385785997863898816778778839887
198 1009070101307091019102103103310491051061063106910710872231103922710911093229241097111711232511292571926326928113132834011511534092934211634431733449411811872743345746137346346734748750349475095211935233751201213953535412754755756356958757601499196076179641976439647965359379765966199676836769809821577397782398277829838385385785997863898816778778839887
199 10090701013070910191021031033104910510610631069107108710911039223110932271097111711232292411292511313257192632692811511532834011634092934211811872743173344334494119345746137346346734748750349475095212012139523375121754127547557563535695358757601499196076179641976439647965359379765966199676836769809821577397782398277829838385385785997863898816778778839887
200 100907010130709101910210310331049105106106310691071087109109311039110971117112322711292292411313251151153257192632692811632834011811872740929342119344317334494120121373457433461394634673474875034947509521217512233752353541275475575635695358757601499196076179641976439647965359379765966199676836769809821577397782398277829838385385785997863898816778778839887

码

这是一个Python 3脚本，可以一遍又一遍地调用Concorde求解器，直到构造解决方案为止。

Concorde免费供学术使用。您可以下载使用其自己的线性编程软件包QSopt构建的Concorde可执行二进制文件，或者如果您以某种方式拥有IBM CPLEX的许可证，则可以从源代码构建Concorde以使用CPLEX。

#!/usr/bin/env python3
'''
Find prime containment numbers (OEIS A054261) using the Concorde
TSP solver.

The n-th prime containment number is the smallest natural number
which, when written in decimal, contains the first n primes.
'''

import argparse
import itertools
import os
import sys
import subprocess
import tempfile

def join_strings(a, b):
  '''Shortest string that starts with a and ends with b.'''
  for overlap in range(min(len(a), len(b)), 0, - 1):
    if a[-overlap:] == b[:overlap]:
      return a + b[overlap:]
  return a + b

def is_prime(n):
  if n < 2:
    return False
  d = 2
  while d*d <= n:
    if n % d == 0:
      return False
    d += 1
  return True

def prime_list_reduced(n):
  '''First n primes, with primes that are substrings of other
     primes removed.'''
  primes = []
  p = 2
  while len(primes) < n:
    if is_prime(p):
      primes.append(p)
    p += 1

  reduced = []
  for p in primes:
    if all(p == q or str(p) not in str(q) for q in primes):
      reduced.append(p)
  return reduced

# w_med is an offset for actual weights
# (we use zero as a dummy weight when splitting nodes)
w_med = 10**4
# w_big blocks edges from being taken
w_big = 10**8

def gen_tsplib(prefix, strs, start_candidates):
  '''Generate TSP formulation in TSPLIB format.

     Returns a TSPLIB format string that encodes the length of the
     shortest string starting with 'prefix' and containing all 'strs'.

     start_candidates is the set of strings that solution paths are
     allowed to start with.
     '''
  N = len(strs)

  # Concorde only supports symmetric TSPs. Therefore we encode the
  # asymmetric TSP instances by doubling each node.
  node_in = lambda i: 2*i
  node_out = lambda i: node_in(i) + 1
  # 2*(N+1) nodes because we add an artificial node with index N
  # for the start/end of the tour. This node is also doubled.
  num_nodes = 2*(N+1)

  # Ensure special offsets are big enough
  assert w_med > len(prefix) + sum(map(len, strs))
  assert w_big > w_med * num_nodes

  weight = [[w_big] * num_nodes for _ in range(num_nodes)]
  def edge(src, dest, w):
    weight[node_out(src)][node_in(dest)] = w
    weight[node_in(dest)][node_out(src)] = w

  # link every incoming node with the matching outgoing node
  for i in range(N+1):
    weight[node_in(i)][node_out(i)] = 0
    weight[node_out(i)][node_in(i)] = 0

  for i, p in enumerate(strs):
    if p in start_candidates:
      prefix_w = len(join_strings(prefix, p))
      # Initial length
      edge(N, i, w_med + prefix_w)
    else:
      edge(N, i, w_big)
    # Link every str to the end to allow closed tours
    edge(i, N, w_med)

  for i, p in enumerate(strs):
    for j, q in enumerate(strs):
      if i != j:
        w = len(join_strings(p, q)) - len(p)
        edge(i, j, w_med + w)

  out = '''NAME: prime-containment-number
TYPE: TSP
DIMENSION: %d
EDGE_WEIGHT_TYPE: EXPLICIT
EDGE_WEIGHT_FORMAT: FULL_MATRIX
EDGE_WEIGHT_SECTION
''' % num_nodes

  out += '\n'.join(
    ' '.join(str(w) for w in row)
    for row in weight
  ) + '\n'

  out += 'EOF\n'
  return out

def parse_tour_soln(prefix, strs, text):
  '''This constructs the solution from Concorde's 'tour' output format.
     The format simply consists of a permutation of the graph nodes.'''
  N = len(strs)
  node_in = lambda i: 2*i
  node_out = lambda i: node_in(i) + 1
  nums = list(map(int, text.split()))

  # The file starts with the number of nodes
  assert nums[0] == 2*(N+1)
  nums = nums[1:]

  # Then it should list a permutation of all nodes
  assert len(nums) == 2*(N+1)

  # Find and remove the artificial starting point
  start = nums.index(node_out(N))
  nums = nums[start+1:] + nums[:start]
  # Also find and remove the end point
  if nums[-1] == node_in(N):
    nums = nums[:-1]
  elif nums[0] == node_in(N):
    # Tour printed in reverse order
    nums = reversed(nums[1:])
  else:
    assert False, 'bad TSP tour'
  soln = prefix
  for i in nums:
    # each prime appears in two adjacent nodes, pick one arbitrarily
    if i % 2 == 0:
      soln = join_strings(soln, strs[i // 2])
  return soln

def scs_length(prefix, strs, start_candidates, concorde_path, concorde_verbose):
  '''Find length of shortest containing string using one call to Concorde.'''
  # Concorde's small-input solver CCHeldKarp, tends to fail with the
  # cryptic error message 'edge too long'. Brute force instead
  if len(strs) <= 5:
    best = len(prefix) + sum(map(len, strs))
    for perm in itertools.permutations(range(len(strs))):
      if perm and strs[perm[0]] not in start_candidates:
        continue
      soln = prefix
      for i in perm:
        soln = join_strings(soln, strs[i])
      best = min(best, len(soln))
    return best

  with tempfile.TemporaryDirectory() as tempdir:
    concorde_path = os.path.join(os.getcwd(), concorde_path)
    with open(os.path.join(tempdir, 'prime.tsplib'), 'w') as f:
      f.write(gen_tsplib(prefix, strs, start_candidates))

    if concorde_verbose:
      subprocess.check_call([concorde_path, os.path.join(tempdir, 'prime.tsplib')],
                            cwd=tempdir)
    else:
      try:
        subprocess.check_output([concorde_path, os.path.join(tempdir, 'prime.tsplib')],
                                cwd=tempdir, stderr=subprocess.STDOUT)
      except subprocess.CalledProcessError as e:
        print('Concorde exited with error code %d\nOutput log:\n%s' %
              (e.returncode, e.stdout.decode('utf-8', errors='ignore')),
              file=sys.stderr)
        raise

    with open(os.path.join(tempdir, 'prime.sol'), 'r') as f:
      soln = parse_tour_soln(prefix, strs, f.read())
    return len(soln)

# Cache results from previous N's
pcn_solve_cache = {} # (prefix fragment, strs) -> soln

def pcn(n, concorde_path, concorde_verbose):
  '''Find smallest prime containment number for first n primes.'''
  strs = list(map(str, prime_list_reduced(n)))
  target_length = scs_length('', strs, strs, concorde_path, concorde_verbose)

  def solve(prefix, strs, target_length):
    if not strs:
      return prefix

    # Extract part of prefix that is relevant to cache
    prefix_fragment = ''
    for s in strs:
      next_prefix = join_strings(prefix, s)
      overlap = len(prefix) + len(s) - len(next_prefix)
      fragment = prefix[len(prefix) - overlap:]
      if len(fragment) > len(prefix_fragment):
        prefix_fragment = fragment
    fixed_prefix = prefix[:len(prefix) - len(prefix_fragment)]
    assert fixed_prefix + prefix_fragment == prefix

    cache_key = (prefix_fragment, tuple(strs))
    if cache_key in pcn_solve_cache:
      return fixed_prefix + pcn_solve_cache[cache_key]

    # Not in cache, we need to calculate it.
    soln = None

    # Try strings in ascending order until scs_length reports a
    # solution with equal length. That string will be the
    # lexicographically smallest extension of our solution.
    next_prefixes = sorted((join_strings(prefix, s), s)
                           for s in strs)

    # Try first string -- often works
    next_prefix, _ = next_prefixes[0]
    next_prefixes = next_prefixes[1:]
    next_strs = [s for s in strs if s not in next_prefix]
    next_length = scs_length(next_prefix, next_strs, next_strs,
                             concorde_path, concorde_verbose)
    if next_length == target_length:
      soln = solve(next_prefix, next_strs, next_length)
    else:
      # If not, do a weighted binary search on remaining strings
      while len(next_prefixes) > 1:
        split = (len(next_prefixes) + 2) // 3
        group = next_prefixes[:split]
        group_length = scs_length(prefix, strs, [s for _, s in group],
                                  concorde_path, concorde_verbose)
        if group_length == target_length:
          next_prefixes = group
        else:
          next_prefixes = next_prefixes[split:]
      if next_prefixes:
        next_prefix, _ = next_prefixes[0]
        next_strs = [s for s in strs if s not in next_prefix]
        check = True
        # Uncomment if paranoid
        #next_length = scs_length(next_prefix, next_strs, next_strs,
        #                         concorde_path, concorde_verbose)
        #check = (next_length == target_length)
        if check:
          soln = solve(next_prefix, next_strs, target_length)

    assert soln is not None, (
      'solve failed! prefix=%r, strs=%r, target_length=%d' %
      (prefix, strs, target_length))

    pcn_solve_cache[cache_key] = soln[len(fixed_prefix):]
    return soln

  return solve('', strs, target_length)

parser = argparse.ArgumentParser()
parser.add_argument('--concorde', type=str, default='concorde',
                    help='path to Concorde binary')
parser.add_argument('--verbose', action='store_true',
                    help='dump all Concorde output')
parser.add_argument('--start', type=int, metavar='N', default=1,
                    help='start at this N')
parser.add_argument('--end', type=int, metavar='N', default=1000000,
                    help='stop after this N')
parser.add_argument('--one', type=int, metavar='N',
                    help='solve for a single N and exit')

def main():
  opts = parser.parse_args(sys.argv[1:])

  if opts.one is not None:
    opts.start = opts.one
    opts.end = opts.one

  prev_soln = ''
  for n in range(opts.start, opts.end+1):
    primes = map(str, prime_list_reduced(n))
    if all(p in prev_soln for p in primes):
      soln = prev_soln
    else:
      soln = pcn(n, opts.concorde, opts.verbose)

    print('%d %s' % (n, soln))
    sys.stdout.flush()
    prev_soln = soln

if __name__ == '__main__':
  main()