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# [擂台] 1-9各用若干次，最多可组成多少个素数？

 k=3应该可以达到14，而k=4我觉得应该可以达到17。

 同意k = 3的 但为什么说k = 4，是17个呢？

楼主| 发表于 2008-10-27 07:44:32 | 显示全部楼层
 这道题是我自编的， 感觉用计算机搜索最大的难点在于设计良好的数据结构和高效的算法。

 有10000内素数表足够了

 需要找除个位数字外，其他位是2，4，5，6，8的素数

 这种数字是 125 * 4 = 500个， 素数应该不多

 而且优先使用位数低的数.k=3和k=4完全可以手工构造 k=3我找到了一个14个数的: 2,3,5,7,29,41,47,53,61,67,83,89,541,6829

 k=4时18还可以达的到,不过看来要借助一下计算机,共199个解: 2,3,5,7,29,41,43,47,53,59,61,67,83,89,421,661,827,859 2,3,5,7,29,41,43,47,53,59,61,67,83,89,421,661,857,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,421,661,587,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,521,641,887,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,521,661,487,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,521,881,647,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,521,881,467,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,821,821,647,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,821,821,647,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,821,821,467,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,821,821,467,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,241,661,827,859 2,3,5,7,29,41,43,47,53,59,61,67,83,89,241,661,857,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,241,661,587,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,541,661,827,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,541,661,887,229 2,3,5,7,29,41,43,47,53,59,61,67,83,89,641,821,827,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,641,821,827,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,641,821,857,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,641,821,587,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,641,881,227,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,641,881,227,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,641,881,257,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,251,641,887,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,251,461,887,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,251,661,487,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,251,881,647,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,251,881,467,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,461,521,887,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,461,821,827,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,461,821,827,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,461,821,857,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,461,821,587,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,461,881,227,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,461,881,227,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,461,881,257,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,661,821,547,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,661,821,457,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,661,881,547,229 2,3,5,7,29,41,43,47,53,59,61,67,83,89,661,881,457,229 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,821,647,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,821,647,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,821,467,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,821,467,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,641,827,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,641,827,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,641,857,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,641,587,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,461,827,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,461,827,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,461,857,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,461,587,269 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,661,547,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,661,457,829 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,281,647,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,281,647,569 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,281,467,659 2,3,5,7,29,41,43,47,53,59,61,67,83,89,281,281,467,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,421,641,887,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,421,641,887,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,421,461,887,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,421,461,887,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,421,661,487,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,421,881,647,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,421,881,647,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,421,881,467,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,421,881,467,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,521,661,887,449 2,3,5,7,23,29,41,47,53,59,61,67,83,89,241,641,887,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,241,641,887,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,241,461,887,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,241,461,887,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,241,661,487,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,241,881,647,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,241,881,647,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,241,881,467,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,241,881,467,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,541,641,887,269 2,3,5,7,23,29,41,47,53,59,61,67,83,89,541,661,487,829 2,3,5,7,23,29,41,47,53,59,61,67,83,89,541,881,647,269 2,3,5,7,23,29,41,47,53,59,61,67,83,89,541,881,467,269 2,3,5,7,23,29,41,47,53,59,61,67,83,89,641,821,647,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,641,821,467,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,641,821,487,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,641,821,487,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,641,641,827,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,641,641,857,829 2,3,5,7,23,29,41,47,53,59,61,67,83,89,641,641,587,829 2,3,5,7,23,29,41,47,53,59,61,67,83,89,641,881,547,269 2,3,5,7,23,29,41,47,53,59,61,67,83,89,641,881,457,269 2,3,5,7,23,29,41,47,53,59,61,67,83,89,251,661,887,449 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,821,647,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,821,467,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,821,487,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,821,487,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,541,887,269 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,641,827,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,641,857,829 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,641,587,829 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,461,827,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,461,857,829 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,461,587,829 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,881,547,269 2,3,5,7,23,29,41,47,53,59,61,67,83,89,461,881,457,269 2,3,5,7,23,29,41,47,53,59,61,67,83,89,661,821,857,449 2,3,5,7,23,29,41,47,53,59,61,67,83,89,661,821,587,449 2,3,5,7,23,29,41,47,53,59,61,67,83,89,661,881,257,449 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,641,647,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,641,467,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,641,487,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,641,487,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,461,647,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,461,467,859 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,461,487,659 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,461,487,569 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,661,857,449 2,3,5,7,23,29,41,47,53,59,61,67,83,89,281,661,587,449 2,3,5,7,23,29,41,43,47,59,61,67,83,89,421,661,857,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,421,661,587,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,521,641,887,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,521,641,887,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,521,661,487,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,521,881,647,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,521,881,647,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,521,881,467,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,521,881,467,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,241,661,857,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,241,661,587,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,541,661,827,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,541,661,857,829 2,3,5,7,23,29,41,43,47,59,61,67,83,89,541,661,587,829 2,3,5,7,23,29,41,43,47,59,61,67,83,89,641,821,857,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,641,821,857,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,641,821,587,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,641,821,587,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,641,881,257,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,641,881,257,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,641,881,557,269 2,3,5,7,23,29,41,43,47,59,61,67,83,89,251,641,887,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,251,641,887,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,251,461,887,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,251,461,887,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,251,661,487,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,251,881,647,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,251,881,647,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,251,881,467,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,251,881,467,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,461,521,887,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,461,521,887,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,461,821,857,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,461,821,857,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,461,821,587,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,461,821,587,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,461,881,257,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,461,881,257,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,461,881,557,269 2,3,5,7,23,29,41,43,47,59,61,67,83,89,661,821,547,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,661,821,457,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,641,857,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,641,857,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,641,587,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,641,587,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,461,857,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,461,857,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,461,587,659 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,461,587,569 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,661,547,859 2,3,5,7,23,29,41,43,47,59,61,67,83,89,281,661,457,859 2,3,5,7,23,29,41,43,47,53,59,61,67,89,421,661,887,859 2,3,5,7,23,29,41,43,47,53,59,61,67,89,821,881,647,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,821,881,647,569 2,3,5,7,23,29,41,43,47,53,59,61,67,89,821,881,467,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,821,881,467,569 2,3,5,7,23,29,41,43,47,53,59,61,67,89,241,661,887,859 2,3,5,7,23,29,41,43,47,53,59,61,67,89,541,661,887,829 2,3,5,7,23,29,41,43,47,53,59,61,67,89,641,821,887,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,641,821,887,569 2,3,5,7,23,29,41,43,47,53,59,61,67,89,641,881,827,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,641,881,827,569 2,3,5,7,23,29,41,43,47,53,59,61,67,89,641,881,857,269 2,3,5,7,23,29,41,43,47,53,59,61,67,89,641,881,587,269 2,3,5,7,23,29,41,43,47,53,59,61,67,89,461,821,887,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,461,821,887,569 2,3,5,7,23,29,41,43,47,53,59,61,67,89,461,881,827,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,461,881,827,569 2,3,5,7,23,29,41,43,47,53,59,61,67,89,461,881,857,269 2,3,5,7,23,29,41,43,47,53,59,61,67,89,461,881,587,269 2,3,5,7,23,29,41,43,47,53,59,61,67,89,661,821,487,859 2,3,5,7,23,29,41,43,47,53,59,61,67,89,661,881,547,829 2,3,5,7,23,29,41,43,47,53,59,61,67,89,661,881,457,829 2,3,5,7,23,29,41,43,47,53,59,61,67,89,281,641,887,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,281,641,887,569 2,3,5,7,23,29,41,43,47,53,59,61,67,89,281,461,887,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,281,461,887,569 2,3,5,7,23,29,41,43,47,53,59,61,67,89,281,661,487,859 2,3,5,7,23,29,41,43,47,53,59,61,67,89,281,881,647,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,281,881,647,569 2,3,5,7,23,29,41,43,47,53,59,61,67,89,281,881,467,659 2,3,5,7,23,29,41,43,47,53,59,61,67,89,281,881,467,569 Total 199 solutions

### 评分

gxqcn + 1 + 1 厉害！

 大k值是否能通过小k值生成结果

 本帖最后由 数论爱好者 于 2021-9-10 10:02 编辑 换个思路研究一下,组合数学我不行 在给定范围内,组成的所有素数,去掉含0组成的素数,剩下的素数全部是由1-9组成的素数,1-9重复的次数各不相同. 在x=10^2,有25个素数,25个素数都没有0出现,所以1-9组成的素数有25个 在x=10^3,有168个素数,有15个含0组成的素数,所以1-9组成的不含0素数有168-15=153个 在x=10^5,有9592个素数,由excel2007统计9592个素数:查找全部0,共计2470个单元格被找到,一个单元格代表一个素数. 所以10^5以内,共有含0素数=2470个 含1素数=4917个 含2素数=3357个 含3素数=4877个 含4素数=3231个 含5素数=3282个 含6素数=3225个 含7素数=4819个 含8素数=3217个 含9素数=4826个 含0素数最少 在10^5以内,由1-9组成的不含0素数有9592-2470=7122个 那么在x=10^12以内,含0素数有多少个?

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