数字串的通项公式

王守恩

northwolves 发表于 2024-12-9 18:32
{420, 1015, 1610, 2205, 2800, 3395, 3990, 4585, 5180}

如果可以瞎猜的话。
{420, 1015, 1610, 2205, 2800, 3395, 3990, 4585, 5180, 5775, 6370, 6965, 7560, 8155, 8750, 9345, 9940}

9^16,  积的最高位数字=1。
99^161,  积的最高位数字=1。
999^1609,  积的最高位数字=1。
......
又该怎样瞎猜？

王守恩

无端感觉这些数字串都应该跟 "Ceiling[n/(1 - Log10@9)]" 有关系。我们应该充分利用。譬如：
Select[Range[100000], First@IntegerDigits[99^#] == 9 &]
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 917, 918, 919, 920, 921,
922, 923, 924, 925, 926, 1146, 1147, 1148, 1149, 1150, 1151, 1152, 1153, 1154, 1155, 1156, 1375, 1376, 1377, 1378, 1379, 1380, 1381, 1382, 1383, 1384, 1385, 1604, 1605, 1606, 1607, 1608, 1609, 1610,
1611, 1612, 1613, 1614, 1833, 1834, 1835, 1836, 1837, 1838, 1839, 1840, 1841, 1842, 1843, 2062, 2063, 2064, 2065, 2066, 2067, 2068, 2069, 2070, 2071, 2072, 2292, 2293, 2294, 2295, 2296, 2297, 2298,
把节点拖出来。
{230, 459, 688, 917, 1146, 1375, 1604, 1833, 2062, 2292, 2521, 2750, 2979, 3208, 3437, 3666, 3895, 4124, 4354, 4583, 4812, 5041, 5270, 5499, 5728, 5957, 6186, 6415, 6645, 6874, 7103, 7332, 7561, 7790,
8019, 8248, 8477, 8707, 8936, 9165, 9394, 9623, 9852, 10081, 10310, 10539, 10768, 10998, 11227, 11456, 11685, 11914, 12143, 12372, 12601, 12830, 13060, 13289, 13518, 13747, 13976, 14205, 14434, ...
Table[Ceiling[(10 Log[10]/Log[10/9] + 897 Log[10]/(89 Log[9])) n], {n, 436}]
问：1,有错的吗？2,还可以精炼？
症结在于对——A135391——没理解透。
Table[N[Ceiling[n*Log[10]/Log[10/9]]*Log[10/9]/(n*Log[10]), 20], {n,19999999999999000, 19999999999999010}]

王守恩

99*99*99*99*99*99*......,  积的最高位数字=9。

1. Select[Range[100000], First@IntegerDigits[99^#] == 9 &]

复制代码

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 917, 918, 919, 920, 921,
922, 923, 924, 925, 926, 1146, 1147, 1148, 1149, 1150, 1151, 1152, 1153, 1154, 1155, 1156, 1375, 1376, 1377, 1378, 1379, 1380, 1381, 1382, 1383, 1384, 1385, 1604, 1605, 1606, 1607, 1608, 1609, 1610,
1611, 1612, 1613, 1614, 1833, 1834, 1835, 1836, 1837, 1838, 1839, 1840, 1841, 1842, 1843, 2062, 2063, 2064, 2065, 2066, 2067, 2068, 2069, 2070, 2071, 2072, 2292, 2293, 2294, 2295, 2296, 2297, 2298,
2299, 2300, 2301, 2521, 2522, 2523, 2524, 2525, 2526, 2527, 2528, 2529, 2530, 2750, 2751, 2752, 2753, 2754, 2755, 2756, 2757, 2758, 2759, 2979, 2980, 2981, 2982, 2983, 2984, 2985, 2986, 2987, 2988,
3208, 3209, 3210, 3211, 3212, 3213, 3214, 3215, 3216, 3217, 3437, 3438, 3439, 3440, 3441, 3442, 3443, 3444, 3445, 3446, 3447, 3666, 3667, 3668, 3669, 3670, 3671, 3672, 3673, 3674, 3675, 3676, 3895,
3896, 3897, 3898, 3899, 3900, 3901, 3902, 3903, 3904, 3905, 4124, 4125, 4126, 4127, 4128, 4129, 4130, 4131, 4132, 4133, 4134, 4354, 4355, 4356, 4357, 4358, 4359, 4360, 4361, 4362, 4363, 4583, 4584,
上面是正确的答案。下面是错误的。
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 917, 918, 919, 920, 921,
922, 923, 924, 925, 926, 1146, 1147, 1148, 1149, 1150, 1151, 1152, 1153, 1154, 1155,          1375, 1376, 1377, 1378, 1379, 1380, 1381, 1382, 1383, 1384,          1604, 1605, 1606, 1607, 1608, 1609, 1610,
1611, 1612, 1613,          1833, 1834, 1835, 1836, 1837, 1838, 1839, 1840, 1841, 1842,                   2063, 2064, 2065, 2066, 2067, 2068, 2069, 2070, 2071, 2072, 2292, 2293, 2294, 2295, 2296, 2297, 2298,
2299, 2300, 2301, 2521, 2522, 2523, 2524, 2525, 2526, 2527, 2528, 2529, 2530, 2750, 2751, 2752, 2753, 2754, 2755, 2756, 2757, 2758, 2759, 2979, 2980, 2981, 2982, 2983, 2984, 2985, 2986, 2987, 2988,
3208, 3209, 3210, 3211, 3212, 3213, 3214, 3215, 3216, 3217, 3437, 3438, 3439, 3440, 3441, 3442, 3443, 3444, 3445, 3446,          3666, 3667, 3668, 3669, 3670, 3671, 3672, 3673, 3674, 3675,          3895,
3896, 3897, 3898, 3899, 3900, 3901, 3902, 3903, 3904,          4124, 4125, 4126, 4127, 4128, 4129, 4130, 4131, 4132, 4133,          4354, 4355, 4356, 4357, 4358, 4359, 4360, 4361, 4362, 4363, 4583, 4584,

1. Flatten@Table[1 + Ceiling[((10 Log[10])/Log[10/9] + (897 Log[10])/(89 Log[9]) - 1/(n + 1)) n] + b, {n, 0, 43}, {b, 0, 9}]

复制代码

王守恩

雷劈数——卡普利加数——Kaprekar 数。

{1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967,
351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 627615, 643357, 648648, 670033, 681318, 791505, 812890, 818181, 851851, 857143, 961038, 994708, 999999,
4444444, 4927941, 5072059, 5479453,5555556, 8161912, 9372385, 9999999,11111112, 13641364,16590564, 19273023, 19773073, 24752475, 25252525, 30884184, 36363636, 38883889, 44363341,
44525548, 49995000, 50005000, 55474452, 55636659, 61116111, 63636364, 69115816, 74747475, 75247525, 80226927, 80726977, 83409436, 86358636, 88888888, 91838088, 94520547,99999999,

1. k[a_] := Module[{n = a^2}, MemberQ[Plus @@ # & /@ Select[Table[{Floor[n/10^j], FractionalPart[n/10^j] 10^j}, {j, IntegerLength[n] - 1}], #[[2]] != 0 &], a]]; Select[Range[10^9], k]

复制代码

详见——OEIS——A053816。

[code]A053816的公式——Select[Range[540000], Total[FromDigits /@ TakeDrop[IntegerDigits[#^2], Floor[IntegerLength[#^2]/2]]] == # &]——有问题。[code]

王守恩

{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225}
Table[Floor[n/(E^(2/n) - 1)], {n, 50}]

王守恩

A013660——1, 2, 5, 7, 11——只有 5 项——我来续几项。

{1, 2, 5, 7, 11, 14, 19, 24, 30, 36, 42, 50, 58, 66, 75, 85, 95, 106, 118, 130, 142, 156, 169, 184, 199, 214, 230, 247, 264, 282, 300, 319, 339, 359, 380, 401, 423, 446, 469, 492, 517, 542, 567, 593, 620, 647, 674, 703},

1. Table[Floor[n/(Pi^(a/n) - 1)], {a, 9}, {n, a, 50}]

复制代码

王守恩

2^0=1,
2^1=2,
2^5=32,
2^2=4,
2^9=512,
2^6=64,
2^46=70368744177664,
2^3=8,
2^53=9007199254740992,
2^10=10,
2^50=1125899906842624,
2^7=128,
......
{0, 1, 5, 2, 9, 6, 46, 3, 53, 10, 50, 7, 17, 47, 77, 4, 34, 54, 84, 11, 31, 51, 61, 81, 8, 18, 38, 48, 68, 78, 98, 5, 25, 35, 45, 55, 75, 85, 95, 12, 22, 32, 42, 145, 52, 62, 72, 82, 92, 102, 9, 19, 29, 39,

这算式怎么编排？

northwolves

王守恩 发表于 2024-12-13 19:03
2^0=1,
2^1=2,
2^5=32,

1. Table[SelectFirst[Range[0, 2500],
   
2.   Take[IntegerDigits[2^#], IntegerLength@n] == IntegerDigits@n &], {n,
   
3.    900}]

复制代码

{0,1,5,2,9,6,46,3,53,10,50,7,17,47,77,4,34,54,84,11,31,51,61,81,8,18,38,48,68,78,98,5,25,35,45,55,75,85,95,12,22,32,42,145,52,62,72,82,92,102,9,19,29,39,142,49,59,162,69,79,89,192,99,6,16,119,26,36,139,46,149,56,66,169,76,179,86,189,96,199,13,209,23,219,33,229,43,239,146,53,156,63,166,73,176,83,279,186,93,196,103,10,113,20,216,123,30,133,40,236,143,50,246,153,60,163,359,70,173,80,276,183,90,286,193,100,296,7,110,306,17,120,316,27,419,130,37,233,140,47,243,150,346,57,449,160,67,263,170,366,77,469,180,87,283,190,386,97,293,200,396,107,14,210,406,117,24,220,416,127,34,230,426,137,44,240,436,147,343,54,446,157,353,64,260,456,167,74,270,466,177,373,84,280,476,187,383,94,486,197,393,104,300,11,207,403,114,310,21,217,413,124,320,31,227,423,134,330,41,237,433,629,144,536,51,247,443,154,350,61,257,453,164,360,556,71,267,463,174,370,81,277,473,669,184,380,91,287,483,679,194,390,101,297,493,8,204,400,111,307,503,18,214,410,121,317,998,28,224,420,616,131,327,38,719,234,430,141,337,1018,48,244,440,1121,151,347,543,58,254,450,646,161,357,553,68,264,460,1141,171,367,563,78,274,470,1151,181,377,1058,88,284,965,480,191,872,387,98,779,294,490,1171,201,397,1078,108,304,985,15,696,211,407,603,118,314,995,25,221,1387,417,613,128,324,1005,35,716,231,427,1108,138,334,1500,45,1211,241,437,1603,148,829,344,540,55,251,1417,447,643,158,354,1520,65,1231,261,942,457,653,168,364,1530,75,1241,271,952,467,663,178,374,1540,85,1251,281,962,477,673,188,869,384,580,95,291,1457,487,1168,198,1364,394,1075,105,786,301,982,12,693,208,889,404,600,115,796,311,507,22,703,218,899,414,610,125,806,321,1002,32,713,228,909,424,1105,135,816,331,1012,42,1208,238,1404,434,1600,630,145,341,1507,537,52,733,248,929,444,1125,155,1321,351,1517,547,62,743,258,939,454,1135,165,1331,361,1527,557,72,753,268,949,464,1145,175,1341,371,2022,1052,82,1248,278,1444,474,1640,670,185,1351,381,1547,577,92,773,288,969,484,1650,680,195,876,391,1557,587,102,783,298,979,494,9,690,205,886,401,1567,597,112,793,308,1474,504,19,1185,215,1381,411,2062,1092,122,1288,318,1969,999,29,1680,710,225,906,421,1587,617,132,1298,328,1979,1009,39,1205,720,235,916,431,1597,627,142,1308,338,1989,1019,49,1700,730,245,1411,441,2092,1122,152,1803,833,348,1514,544,59,1225,255,1906,936,451,1617,647,162,1328,843,358,1524,554,69,1235,265,1916,946,461,1627,1142,172,1338,853,368,1534,564,79,1245,760,275,1441,471,2122,1152,182,1833,1348,378,2029,1059,89,1740,1255,285,1936,966,481,1647,1162,192,1843,873,388,1554,1069,99,1750,780,295,1946,976,491,1657,1172,202,1853,883,398,2049,1079,594,109,1275,305,1956,986,501,16,1182,697,212,1378,893,408,1574,604,119,1770,800,315,1481,996,511,26,1192,222,1873,1388,418,2069,1584,614,129,1780,810,325,1976,1006,521,36,1202,717,232,1398,913,428,1594,1109,139,1790,1305,335,1986,1501,531,46,1697,1212,242,1893,1408,438,2089,1604,634,149,1800,830,345,1996,1511,541,56,1707,737,252,1903,1418,448,2099,1614,644,159,1810,1325,355,2006,1521,1036,66,1717,1232,747,262,1428,943,458,1624,1139,654,169,1335,850,365,2016,1531,561,76,1727,1242,272,1923,1438,953,468,1634,1149,664,179,1345,860,375,2026,1541,571,86,1737,1252,767,282,1448,963,478,2129,1644,674,189,1840,1355,870,385,1551,1066,581,96,1747,777,292,1943,1458,973,488,1654,1169,684,199,1850,1365,395,2046,1561,1076,591,106,1757,787,302,1953,1468,983,498,13,1664,694,209,1860,1375,890,405,2056,1571,601,116,1767,1282,797,312,1963,1478,508,23,1674,1189,704,219,1870,1385,900,415,2066,1096,611,126,1777,1292,807,322,1973,1488,1003,33,2169,1199,714,229,1880,1395,910,425,2076,1591,1106,621,136,1787,817,332,1983,1498,1013,528,43,1694,1209,724,239,1890,1405,920,435,2086,1601,1116,631,146,1797,827,342,1993,1508,1023,538,53}

northwolves

再贪心: 6*6*6*6*6*6*......, 最高位数字=666666
-----------------------------------------------------------------
{1,118,420,126862,128647,2121356}

王守恩

B(1)=1=1
B(2)=10=1
B(3)=11=2
B(4)=100=1
B(5)=101=2
B(6)=110=2
B(7)=111=3
B(8)=1000=1
B(9)=1001=2
B(10)=1010=2
B(11)=1011=3
B(12)=1100=2
B(13)=1101=3
B(14)=1110=3
B(15)=1111=4
B(16)=10000=1
B(17)=10001=2

{1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4,
5, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4,...}

1. Table[DigitCount[n, 2, 1], {n, 120}]

复制代码

\(\D\exp\bigg(\sum_{n=1}^{\infty}\frac{B(n)}{n(n+1)}\bigg)=\exp\big(\ln(4)\big)=4\)