这样的A有多少个？

northwolves · 发表于 2024-3-12 16:50:18

northwolves 发表于 2024-3-6 17:31
1 9 3
2 13 7
3 19 17

感觉可能性比较小，毕竟候选数的比例>3:1而且还多出一位

王守恩 · 发表于 2024-3-13 08:14:39

谢谢 mathe！这{}里的数还得来几个。目的:  找a(n)。找>33/4的无限数串(应该有)，一直没找到。
A*A=n位数,  A*A各个数位上最大数码和=a(n)。
n    a(n) A
01 009, 3,
02 013, 7,
03 019, 17,{17, 26, 28}
04 031, 83,
05 040, 313,
06 046, 836,{836, 883, 937}
07 054, 3114,
08 063, 8937,{8937, 9417}
09 070, 29614,
10 081, 94863,
11 088, 298327,
12 097, 987917,
13 106, 3162083,
14 112, 9893887,
15 121, 29983327,
16 130, 99477133,{99477133, 99483667}
17 136, 197483417,{197483417, 282753937, 314623583, 315432874}
18 148, 994927133,
19 154, 2983284917,
20 162, 9380293167,9486778167,
21 171, 28105157886,
22 180, 99497231067,
23 187, 244272388937,
24 193, 926174913167,999949483667,
25 205, 3160522105583,
26 211, 9892825177313,
27 220, 29999983333327,
28 229, 89324067192437,
29 235, 314451904109293,
30 244, 943291047332683,
31 253, 2641018364192114,
32 262, 9949874270443813,
33 271, 31622774688331667,
34 277, 83066231922477313,
35 286, 299999998333333327,
36 297, 707106074079263583,
37 301, 2785480209927724417,
38 310, 9429681807356492126,
39 319, 29964628614384477133,
40 331, 94180040294109027313,
41 334, 223584183022838178583,
42 343, 888142995231510436417,
43 355, 2976388751488907738914,
44 360, 8882505274864168010583,
45 367, 24490814584000030724417,
46 378, 99689518004050952477133,
47 388, 312713447088224669275583,
48 396, 893241282627485818275387,
49 ?                   ?
50 ?                   ?
51 ?                   ?
52 ?                   ?
53 ?                   ?
54 ?                   ?
55 ?                   ?
56 ?                   ?
57 ?                   ?
58 ?                   ?
59 487, 314610537013606681884298837387,
60 ?                   ?
61 ?                   ?
62 513, 9984988582817657883693383344833,
63 ?                   ?
......
这{}里的数还得来几个(这些数还是有些规律的)。我这个电脑只能到17(这奇数位,还是可以利用)。我的电脑算下面,  还是很强大的。

Table[Total[IntegerDigits[n^2]], {k, 999999, 999999}, {n, 3*10^(2 k - 1) - (5*10^k + 19)/3 - 9, 3*10^(2 k - 1) - (5*10^k + 19)/3 + 9}]

复制代码

{{32999929, 29999941, 26999946, 32999926, 29999953, 26999955, 32999941, 29999947, 26999955, 32999956, 29999959, 17999964, 23999953, 20999962, 17999973, 23999968, 20999965, 17999973, 23999965}}

王守恩 · 发表于 2024-3-13 12:37:43

数字万花筒。数码和也是有规律的！！！

Table[Total[IntegerDigits[n^2]], {a, 2, 9}, {k, 10^a - 1, 10^a - 1}, {n, 3*10^(2 k - 1) - (5*10^k + 19)/3 - 9, 3*10^(2 k - 1) - (5*10^k + 19)/3 + 9}]

复制代码

{{3229, 2941, 2646, 3226, 2953, 2655, 3241, 2947, 2655, 3256, 2959, 1764, 2353, 2062, 1773, 2368, 2065, 1773, 2365}},
{{32929, 29941, 26946, 32926, 29953, 26955, 32941, 29947, 26955, 32956, 29959, 17964, 23953, 20962, 17973, 23968, 20965, 17973, 23965}},
{{329929, 299941, 269946, 329926, 299953, 269955, 329941, 299947, 269955, 329956, 299959, 179964, 239953, 209962, 179973, 239968, 209965, 179973, 239965}},
{{3299929, 2999941, 2699946, 3299926, 2999953, 2699955, 3299941, 2999947, 2699955, 3299956, 2999959, 1799964, 2399953, 2099962, 1799973, 2399968, 2099965, 1799973, 2399965}},
{{32999929, 29999941, 26999946, 32999926, 29999953, 26999955, 32999941, 29999947, 26999955, 32999956, 29999959, 17999964, 23999953, 20999962, 17999973, 23999968, 20999965, 17999973, 23999965}}

王守恩 · 发表于 2024-3-13 16:00:36

这样的无限模式还是不太多。

Table[Total[IntegerDigits[n^2]], {a, 2, 9}, {k, 10^a-1, 10^a-1}, {n, (5*10^k+1)/3-20, (5*10^k+1)/3}]

复制代码

{{1174, 1468, 891, 1180, 1462, 882, 1177, 1465, 891, 1183, 1468, 891, 1189, 1480, 891, 1177, 1474, 891, 1192, 1486, 891, 1189, 1489}},
{{11974, 14968, 8991, 11980, 14962, 8982, 11977, 14965, 8991, 11983, 14968, 8991, 11989, 14980, 8991, 11977, 14974, 8991, 11992, 14986, 8991, 11989, 14989}},
{{119974, 149968, 89991, 119980, 149962, 89982, 119977, 149965, 89991, 119983, 149968, 89991, 119989, 149980, 89991, 119977, 149974, 89991, 119992, 149986, 89991, 119989, 149989}},
{{1199974, 1499968, 899991, 1199980, 1499962, 899982, 1199977, 1499965, 899991, 1199983, 1499968, 899991, 1199989, 1499980, 899991, 1199977, 1499974, 899991, 1199992, 1499986, 899991, 1199989, 1499989}},
{{11999974, 14999968, 8999991, 11999980, 14999962, 8999982, 11999977, 14999965, 8999991, 11999983,14999968, 8999991, 11999989, 14999980, 8999991, 11999977, 14999974, 8999991, 11999992, 14999986, 8999991,11999989,14999989}

王守恩 · 发表于 2024-3-14 07:26:08

A*A*A=n位数, A*A*A各个数位上最大数码和=a(n)。
a(01)=8,
a(02)=10,
a(03)=18,
a(04)=28,
a(05)=28,
a(06)=44,
a(07)=46,
a(08)=54,
a(09)=63,
a(10)=73,
a(11)=80,
a(12)=82,
a(13)=98,
......

northwolves · 发表于 2024-3-14 07:59:46

本帖最后由 northwolves 于 2024-3-15 23:32 编辑

王守恩发表于 2024-3-14 07:26
A*A*A=n位数, A*A*A各个数位上最大数码和=a(n)。
a(01)=8,
a(02)=10,

Table[Max@Map[Total@IntegerDigits[#^3]&,Range[Ceiling@CubeRoot[10^(n-1)],CubeRoot[10^n-1]]],{n,20}]

复制代码

{8,10,18,28,28,44,46,54,63,73,80,82,98,100,109,118,125,136,144,154,154,163,172,181,190,190,199,208,208,217,234,235,244,252,261,262,270,280,289,296}

northwolves · 发表于 2024-3-14 08:34:04

是否存在某个整数k,满足k^2(m位）是m位平方数数字和最大的一个，并且k^3(n位）是n位立方数数字和最大的一个？

王守恩 · 发表于 2024-3-14 09:59:46

northwolves 发表于 2024-3-14 07:59
{8, 10, 18, 28, 28, 44, 46, 54, 63, 73, 80, 82, 98, 100, 109, 118, 125, 136, 144, 154}

A^k=n位数, A^k各个数位上最大数码和=a(n)。随着k增大, a(n)<a(n-1)还是有的。

Table[Max@Table[Total@IntegerDigits[A^k], {A, Floor[Power[10^(n - 1), (k)^-1]], Power[10^n - 1, (k)^-1]}], {k, 2, 9}, {n, 12 + 2k}]

复制代码

{9, 13, 19, 31, 40, 46, 54, 63, 70, 81, 88, 97, 106, 112, 121, 130},
{8, 10, 18, 28, 28, 44, 46, 54, 63, 73, 80, 82, 098, 100, 109, 118, 125, 136},
{1, 09, 13, 19, 25, 37, 43, 52, 55, 70, 76, 79, 085, 099, 103, 108, 118, 127, 135, 142},
{1, 05, 09, 27, 27, 36, 45, 46, 52, 63, 72, 80, 089, 090, 099, 104, 108, 119, 126, 143, 137, 152},
{1, 10, 18, 19, 27, 28, 45, 37, 46, 64, 64, 81, 082, 082, 091, 100, 100, 118, 117, 126, 136, 136, 154, 154},
{1, 01, 11, 18, 23, 36, 45, 40, 55, 58, 62, 63, 072, 081, 090, 097, 107, 115, 121, 126, 128, 144, 148, 145, 154, 162},
{1, 01, 13, 18, 25, 25, 36, 37, 54, 58, 61, 64, 076, 073, 076, 090, 099, 109, 108, 118, 133, 133, 136, 142, 162, 162, 171, 162},
{1, 01, 08, 08, 27, 27, 26, 36, 45, 53, 64, 73, 081, 081, 082, 098, 099, 091, 118, 118, 116, 118, 135, 145, 154, 161, 163, 163, 170, 181}}

王守恩 · 发表于 2024-3-15 08:36:10

A^k=n位数, A^k各个数位上最大数码和=a(n)。随着k增大, a(n)<a(n-1)还是有的。

Table[Max@Table[Total@IntegerDigits[A^k], {A, Floor[Power[10^(n - 1), (k)^-1]], Power[10^n - 1, (k)^-1]}], {k, 1, 9}, {n, 24 + 2k}]

复制代码

{9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234}
{9, 13, 19, 31, 40, 46, 54, 63, 70, 81, 88, 097, 106, 112, 121, 130, 136, 148, 154, 162, 171, 180, 187, 193, 205, 211, 220, 229}
{8, 10, 18, 28, 28, 44, 46, 54, 63, 73, 80, 082, 098, 100, 109, 118, 125, 136, 144, 154, 154, 163, 172, 181, 190}
{1, 09, 13, 19, 25, 37, 43, 52, 55, 70, 76, 079, 085, 099, 103, 108, 118, 127, 135, 142, 144, 153, 171, 171, 178, 181, 189, 198, 205, 211, 220, 232}
{1, 05, 09, 27, 27, 36, 45, 46, 52, 63, 72, 080, 089, 090, 099, 104, 108, 119, 126, 143, 137, 152, 157, 162, 175, 180, 182, 189, 198, 208, 209, 216, 225, 234}
{1, 10, 18, 19, 27, 28, 45, 37, 46, 64, 64, 081, 082, 082, 091, 100, 100, 118, 117, 126, 136, 136, 154, 154, 163, 163, 172, 181, 181, 190, 199, 208, 217, 226, 235, 235}
{1, 01, 11, 18, 23, 36, 45, 40, 55, 58, 62, 063, 072, 081, 090, 097, 107, 115, 121, 126, 128, 144, 148, 145, 154, 162, 180, 176, 200, 189, 200, 198, 207, 217, 224, 242, 241, 243}
{1, 01, 13, 18, 25, 25, 36, 37, 54, 58, 61, 064, 076, 073, 076, 090, 099, 109, 108, 118, 133, 133, 136, 142, 162, 162, 171, 162, 175, 180, 181, 193, 198, 208, 225, 226, 229, 235, 235, 247}
{1, 01, 08, 08, 27, 27, 26, 36, 45, 53, 64, 073, 081, 081, 082, 098, 099, 091, 118, 118, 116, 118, 135, 145, 154, 161, 163, 163, 170, 181, 189, 189, 207, 198, 216, 224, 226, 226, 234, 253, 244, 260}
......
k=2是最困难的一个(k=1虽然困难,但有规律)。后面的好像容易找一些。譬如：
{1, 01, 08, 08, 27, 27, 26, 36, 45, 53, 64, 073, 081, 081, 082, 098, 099, 091, 118, 118, 116, 118, 135, 145, 154, 161, 163, 163, 170, 181, 189, 189, 207, 198, 216, 224, 226, 226, 234, 253,
244, 260, 262, 271, 271, 289, 280, 296, 307, 306, 316, 315, 333, 333, 343, 351, 369, 369, 378, 378, 378, 379, 395, 397, 413, 415, 424, 424, 432, 433, 442, 459}

王守恩 · 发表于 2024-3-16 09:04:27

再来几个？目的：k=2。
A^9各个数位上的数码和=9*3。A最小=3*01, (3*01)^9=19683,
A^9各个数位上的数码和=9*4。A最小=3*02, (3*02)^9=10077696,
A^9各个数位上的数码和=9*5。A最小=3*03, (3*03)^9=387420489,
A^9各个数位上的数码和=9*6。A最小=3*05, (3*05)^9=38443359375,
A^9各个数位上的数码和=9*7。A最小=3*06, (3*06)^9=198359290368,
A^9各个数位上的数码和=9*8。A最小=3*12, (3*12)^9=101559956668416,
A^9各个数位上的数码和=9*9。A最小=3*09, (3*09)^9=7625597484987,
......
得到这样一串数: {1, 2, 3, 5, 6, 12, 9, 32, 24, 53, 79, 127, 116, 231, 185, 492, 674, 856, 831, 1606, 1482, 2482, 4305, 5823, 10409, 13933, 19749, 32086, ...

n = 3; lst = {}; For[k = 0, k < 10^8, k += 3, If[Total[IntegerDigits[k^9]] == 9*n, AppendTo[lst, k]; n++]]; lst

复制代码

{3, 6, 9, 15, 18, 36, 66, 96, 129, 159, 237, 381, 438, 693, 819, 1476, 2022, 2568, 3303, 4818, 6939, 7446, 12915, 17469, 31227, 41799, 59247, 96258, 125064, 211026,
232614, 305226, 436806, 706059, 765156, 1219551, 1350756, 3052353, 5531517, 5878992, 8906538, 13509534, 18618906, 23264568, 36667776, 44495439, 78075918}

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[灌水] 这样的A有多少个？

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