这样的A有多少个？

王守恩 · 发表于 2024-2-23 18:41

王守恩发表于 2024-2-23 17:00
A=十位数，由0,1,2,3,4,5,6,7,8,9组成，
A^2=二十位数，由0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9组成。 ...

4432个？！太多了吧？OEIS没有的！
我好不容易找了1个：7839425610^2=61456593894723872100

王守恩 · 发表于 2024-2-24 09:00

Treenewbee 发表于 2024-2-23 15:51
46楼比值与最小n:
{8.230, 119}, {8.231, 125}, {8.232, 132}, {8.233, 140}, {8.234, 148},{8.235, 158 ...

2976388751488907738914^2
=8858889999989698989999979889997999989899396=355/43

312713447088224669275583^2
=97789699989799889886988697997987998989999989889=388/47

893241282627485818275387^2
=797879988989995997899989877988999997998969999769=396/48

上面3个数都满足, 比值(数码和: 数位)≥33/4。
这样的数，比值(数码和: 数位)≥33/4。能来几个(小一点的应该有吧)？谢谢！

王守恩 · 发表于 2024-2-24 09:52

有兴趣的网友可浏览一下。可惜漏了主帖1#——13#  。

   A369955——2024年2月19日
3, 63, 3114, 8937, 94863, 5477133, 82395381, 706399164, 9380293167,
99497231067, 4472135831667, 62441868958167, 836594274358167,
9983486364492063, 435866837461509417, 707106074079263583,
77453069648658793167, 754718284918279954614, 8882505274864168010583,

   A369953——2024年2月10日
0, 3, 24, 63, 264, 1374, 3114, 8937, 60663, 94863, 545793, 1989417, 5477133,
20736417, 82395387, 260191833, 706399164, 2428989417, 9380293167,
28105157886, 99497231067, 538479339417, 1974763271886, 4472135831667,
14106593458167, 62441868958167, 244744764757083, 836594274358167,

   A369956——2024年2月06日
1, 101, 11, 2, 149, 32, 4, 12, 3, 8, 106, 16, 7, 103, 13, 108, 24, 17, 1019, 124, 43, 1013,
113, 67, 114, 63, 10024, 1024, 133, 83, 1067, 167, 1044, 264, 314, 10087, 1303, 313,
10093, 1183, 707, 1374, 1333, 836, 10343, 1667, 100264, 10714, 2236, 10386, 3114,

王守恩 · 发表于 2024-2-24 13:37

Treenewbee 发表于 2024-2-23 15:51
46楼比值与最小n:
{8.230, 119}, {8.231, 125}, {8.232, 132}, {8.233, 140}, {8.234, 148},{8.235, 158 ...

这样的数会很多吗？谢谢！

Select[Range[10^7],Floor[Total[IntegerDigits[#^2]]/IntegerLength[#^2]]≥8&]

复制代码

王守恩 · 发表于 2024-2-24 15:02

Treenewbee 发表于 2024-2-23 15:51
46楼比值与最小n:
{8.230, 119}, {8.231, 125}, {8.232, 132}, {8.233, 140}, {8.234, 148},{8.235, 158 ...

这样的数会很多吗？谢谢！

Select[Range[10^7],Floor[Total[IntegerDigits[#^2]]/IntegerLength[#^2]]≥8&]

复制代码

参考OEIS——A360803

王守恩 · 发表于 2024-3-6 08:53

A*A=n位数, A*A各个数位上最大数码和, 下面的数字串(无限)有一个合理的推测。

{"583", "339889"},
{"2827", "7991929"},
{"5983", "35796289"},
{"29827", "889649929"},
{"59833", "3579987889"},
{"298327", "88998998929"},
{"599833", "359799627889"},
{"2998327", "8989964798929"},
{"5998333", "35979998778889"},
{"29983327", "898999897988929"},
{"59998333", "3599799962778889"},
{"299983327", "89989996477988929"},
{"599983333", "359979999877788889"},
{"2999833327", "8998999989779888929"},
{"5999983333", "35999799996277788889"},
{"29999833327", "899989999647779888929"},
{"59999833333", "3599979999987777888889"},
{"299998333327", "89998999998977798888929"},
{"599999833333", "359999799999627777888889"},
{"2999998333327", "8999989999964777798888929"},
{"5999998333333", "35999979999998777778888889"},
{"29999983333327", "899998999999897777988888929"},
{"59999998333333", "3599999799999962777778888889"},
{"299999983333327", "89999989999996477777988888929"},
{"599999983333333", "359999979999999877777788888889"},
{"2999999833333327", "8999998999999989777779888888929"},
{"5999999983333333", "35999999799999996277777788888889"},
{"29999999833333327", "899999989999999647777779888888929"},
{"59999999833333333", "3599999979999999987777777888888889"},
{"299999998333333327", "89999998999999998977777798888888929"},
{"599999999833333333", "359999999799999999627777777888888889"},
{"2999999998333333327", "8999999989999999964777777798888888929"},
{"5999999998333333333", "35999999979999999998777777778888888889"},
{"29999999983333333327", "899999998999999999897777777988888888929"},
{"59999999998333333333", "3599999999799999999962777777778888888889"},
{"299999999983333333327", "89999999989999999996477777777988888888929"},
{"599999999983333333333", "359999999979999999999877777777788888888889"},
{"2999999999833333333327", "8999999998999999999989777777779888888888929"},
{"5999999999983333333333", "35999999999799999999996277777777788888888889"},
{"29999999999833333333327", "899999999989999999999647777777779888888888929"},
{"59999999999833333333333", "3599999999979999999999987777777777888888888889"},
{"299999999998333333333327", "89999999998999999999998977777777798888888888929"},
{"599999999999833333333333", "359999999999799999999999627777777777888888888889"},
{"2999999999998333333333327", "8999999999989999999999964777777777798888888888929"},
{"5999999999998333333333333", "35999999999979999999999998777777777778888888888889"},

FullSimplify@Table[k=(1800*10^(n/2)(3(Sqrt[10]-4)Cos[n*Pi]-8-Sqrt[10])
-5*10^(n/4) (2 (11 Sqrt[10]-20) (10 Cos[(n*Pi/2]-10^(1/4) Sin[(n*Pi/2])+
9Sqrt[10]((10^(1/4)-10)Cos[n*Pi]-(10^(1/4)-(10^(1/4)-10)*I*Sin[n*Pi]+10)))
-4(10+9Cos[n*Pi])(Sqrt[10]-10))/(12(Sqrt[10]-10));{k,k^2},{n,40}]//TableForm

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数码和/数位是这样一串数。
{20/3, 55/7, 49/8, 64/9, 73/10, 8, 41/6, 97/13, 53/7, 121/15, 115/16, 130/17, 139/18, 154/19, 37/5,
163/21, 86/11, 187/23, 181/24, 196/25, 205/26, 220/27, 107/14, 229/29, 119/15,253/31, 247/32,
262/33, 271/34, 286/35, 70/9, 295/37, 8, 319/39, 313/40, 8, 337/42, 352/43,173/22,361/45,185/23,
385/47, 379/48, 394/49, 403/50, 418/51, 103/13, 427/53, 218/27, 41/5, 445/56, 460/57}

Table[( 66 n + 27 Cos[n*Pi] - 33 Cos[2 n*Pi] + 15 ((1 - I) (-I)^n
+ (1 + I) I^n) + 152)/(8 (n + 3)), {n, 3, 54}]

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王守恩 · 发表于 2024-3-12 07:54

谢谢 mathe！
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,
04 031, 83,
05 040, 313,
06 046, 836,
07 054, 3114,
08 063, 8937,
09 070, 29614,
10 081, 94863,
11 088, 298327,
12 097, 987917,
13 106, 3162083,
14 112, 9893887,
15 121, 29983327,
16 130, 99477133,
17 136, 197483417,
18 148, 994927133,
19 154, 2983284917,
20 162, 9380293167,
21 171, 28105157886,
22 180, 99497231067,
23 187, 244272388937,
24 193, 926174913167,
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,
48 397, 893241282627485818275387,
49 ?                   ?
50 ?                   ?
51 ?                   ?
52 ?                   ?
53 ?                   ?
54 ?                   ?
55 ?                   ?
56 ?                   ?
57 ?                   ?
58 ?                   ?
59 487, 314610537013606681884298837387,
60 ?                   ?
61 ?                   ?
62 513, 9984988582817657883693383344833,
63 ?                   ?
......

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