[欣赏]找"好数"。

王守恩

正整数A的数字和=B,  A+B的数字和还是=B。我们叫A为"好数"。譬如：
351的数字和=9,   351+9=360,   360的数字和还是=9。
558的数字和=18, 558+18=576, 576的数字和还是=18。

wlc1

A=2025

wlc1

A是9的倍数，但，反之，未必，

wlc1

A=225
A=225+9=234
A=234+9=243
A=243+9=252
A=252+9=261
A=261+9=270，都可以，，，

王守恩

正整数A的数字和=B,  A+B的数字和还是=B。我们叫A为"好数"。

OEIS——A131418——有个外国人是这样猜想：所有项都可以被 9 整除。- Harvey P. Dale， 2019年3月28日
{9, 18, 27, 36, 45, 54, 63, 72, 81, 108, 117, 126, 135, 144, 153, 162, 171, 207, 216, 225, 234, 243, 252, 261, 279, 306, 315, 324, 333,
342, 351, 369, 378, 405, 414, 423, 432, 441, 459, 468, 477, 504, 513, 522, 531, 549, 558, 567, 576, 603, 612, 621, 639, 648, 657, 666,
675, 702, 711, 729, 738, 747, 756, 765, 774, 801, 819, 828, 837, 846, 855, 864, 873, 891, 909, 918, 927, 936, 945, 954, 963, 972, ......

我们来看另外一串数——OEIS——没有这串数。
正整数A的数字和=11的倍数余9,  A+9的数字和还是=11的倍数余9。我们叫A为"好数"。
{9, 18, 27, 36, 45, 54, 63, 72, 81, 108, 117, 126, 135, 144, 153, 162, 171, 207, 216, 225, 234, 243, 252, 261, 306, 315, 324,
333, 342, 351, 389, 405, 414, 423, 432, 441, 479, 488, 504, 513, 522, 531, 569, 578, 587, 603, 612, 621, 659, 668, 677, 686,
702, 711, 749, 758, 767, 776, 785, 801, 839, 848, 857, 866, 875, 884, 929, 938, 947, 956, 965, 974, 983, ......

王守恩

A的数字和=11的倍数,  A+9的数字和还是=11的倍数。
{29, 38, 47, 56, 65, 74, 83, 119, 128, 137, 146, 155, 164, 173, 182, 209, 218, 227, 236, 245, 254, 263, 272, 281, 308, 317, 326, 335, 344, 353, 362, 371, 407, 416, 425, 434, 443, 452, 461, 506, 515, 524, 533,
542, 551, 589, 605, 614, 623, 632, 641, 679, 688, 704, 713, 722, 731, 769, 778, 787, 803, 812, 821, 859, 868, 877, 886, 902, 911, 949, 958, 967, 976, 985, 1019, 1028, 1037, 1046, 1055, 1064, 1073, 1082, 1109,

A的数字和=11的倍数,  A+7的数字和还是=11的倍数。
{499, 598, 697, 796, 895, 1399, 1498, 1597, 1696, 1795, 1894, 2299, 2398, 2497, 2596, 2695, 2794, 2893, 3199, 3298, 3397, 3496, 3595, 3694, 3793, 4099, 4198, 4297, 4396, 4495, 4594, 4693, 5098, 5197,
5296, 5395, 5494, 5593, 6097, 6196, 6295, 6394, 6493, 7096, 7195, 7294, 7393, 7899, 8095, 8194, 8293, 8799, 8898, 9094, 9193, 9699, 9798, 9897, 10399, 10498, 10597, 10696, 10795, 10894, 11299, 11398,

A的数字和=11的倍数,  A+5的数字和还是=11的倍数。
{6999, 7998, 8997, 15999, 16998, 17997, 18996, 24999, 25998, 26997, 27996, 28995, 33999, 34998, 35997, 36996, 37995, 42999, 43998, 44997, 45996, 46995, 51999, 52998, 53997, 54996, 55995, 60999,
61998, 62997, 63996, 64995, 70998, 71997, 72996, 73995, 80997, 81996, 82995, 90996, 91995, 98999, 105999, 106998, 107997, 108996, 114999, 115998, 116997, 117996, 118995, 123999, 124998, 125997,

A的数字和=11的倍数,  A+3的数字和还是=11的倍数。
{89999, 179999, 189998, 269999, 279998, 289997, 359999, 369998, 379997, 449999, 459998, 469997, 539999, 549998, 559997, 629999, 639998, 649997, 719999, 729998, 739997, 809999, 819998, 829997,
909998, 919997, 1079999, 1089998, 1169999, 1179998, 1189997, 1259999, 1269998, 1279997, 1349999, 1359998, 1369997, 1439999, 1449998, 1459997, 1529999, 1539998, 1549997, 1619999, 1629998, 1639997,

A的数字和=11的倍数,  A+1的数字和还是=11的倍数。
{2899999, 3799999, 4699999, 5599999, 6499999, 7399999, 8299999, 9199999, 11899999, 12799999, 13699999, 14599999, 15499999, 16399999, 17299999, 18199999, 19099999, 20899999, 21799999, 22699999,
23599999, 24499999, 25399999, 26299999, 27199999, 28099999, 30799999, 31699999, 32599999, 33499999, 34399999, 35299999, 36199999, 37099999, 40699999, 41599999, 42499999, 43399999, 44299999, 45199999,

真题来了:  
A的数字和=11的倍数,  A+2的数字和还是=11的倍数。求助:  来几个(2个也行)。谢谢！

王守恩

[欣赏]A+(A+1)的数字和=4。

A=6, 15, 51, 60, 105, 150, 501, 510, 555, 600, 1005, 1050, 1500, 5001, 5010, 5055, 5100, 5505, 5550, 6000, 10005, 10050, 10500, 15000, 50001, 50010, 50055, 50100, 50505, 50550, 51000, 55005, 55050, 55500, 60000,
100005, 100050, 100500, 105000, 150000, 500001, 500010, 500055, 500100, 500505, 500550, 501000, 505005, 505050, 505500, 510000, 550005, 550050, 550500, 555000, 600000, 1000005, 1000050, 1000500, 1005000,
1050000, 1500000, 5000001, 5000010, 5000055, 5000100, 5000505, 5000550, 5001000, 5005005, 5005050, 5005500, 5010000, 5050005, 5050050, 5050500, 5055000, 5100000, 5500005, 5500050, 5500500, 5505000,
5550000, 6000000, 10000005, 10000050, 10000500, 10005000, 10050000, 10500000, 15000000, 50000001, 50000010, 50000055, 50000100, 50000505, 50000550, 50001000, 50005005, 50005050, 50005500, 50010000,
50050005, 50050050, 50050500, 50055000, 50100000, 50500005, 50500050, 50500500, 50505000, 50550000, 51000000, 55000005, 55000050, 55000500, 55005000, 55050000, 55500000, 60000000, 100000005, 100000050}

[欣赏]——漂亮整洁的数字串——OEIS没有——你来给她配个通项？ 谢谢！

王守恩

王守恩 发表于 2024-8-18 12:23
[欣赏]A+(A+1)的数字和=4。

A=6, 15, 51, 60, 105, 150, 501, 510, 555, 600, 1005, 1050, 1500, 5001,  ...

谢谢 northwolves！
{6, 15, 51, 60, 105, 150, 501, 510, 555, 600, 1005, 1050, 1500, 5001, 5010, 5055, 5100, 5505, 5550, 6000, 10005, 10050, 10500, 15000, 50001, 50010, 50055, 50100, 50505, 50550, 51000, 55005, 55050, 55500, 60000,
100005, 100050, 100500, 105000, 150000, 500001, 500010, 500055, 500100, 500505, 500550, 501000, 505005, 505050, 505500, 510000, 550005, 550050, 550500, 555000, 600000, 1000005, 1000050, 1000500, 1005000,
1050000, 1500000, 5000001, 5000010, 5000055, 5000100, 5000505, 5000550, 5001000, 5005005, 5005050, 5005500, 5010000, 5050005, 5050050, 5050500, 5055000, 5100000, 5500005, 5500050, 5500500, 5505000,
5550000, 6000000, 10000005, 10000050, 10000500, 10005000, 10050000, 10500000, 15000000, 50000001, 50000010, 50000055, 50000100, 50000505, 50000550, 50001000, 50005005, 50005050, 50005500, 50010000,
50050005, 50050050, 50050500, 50055000, 50100000, 50500005, 50500050, 50500500, 50505000, 50550000, 51000000, 55000005, 55000050, 55000500, 55005000, 55050000, 55500000, 60000000, 100000005, 100000050}

1. Sort[r = Range@10; Join[6*10^(r - 1), 5*Total /@ Subsets[10^(r - 1), {3}], Flatten@Table[{10^s + 5*10^t, 5*10^s + 10^t}, {s, r}, {t, 0, s - 1}]]]

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每个数 - 1,  还是一串数。OEIS就肯定没有了。

1. Sort[r = Range@10; Join[6*10^(r - 1), 5*Total /@ Subsets[10^(r - 1), {3}], Flatten@Table[{10^s + 5*10^t, 5*10^s + 10^t}, {s, r}, {t, 0, s - 1}]]] - 1

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{5, 14, 50, 59, 104, 149, 500, 509, 554, 599, 1004, 1049, 1499, 5000, 5009, 5054, 5099, 5504, 5549, 5999, 10004, 10049, 10499, 14999, 50000, 50009, 50054, 50099, 50504, 50549, 50999, 55004, 55049, 55499, 59999,
100004, 100049, 100499, 104999, 149999, 500000, 500009, 500054, 500099, 500504, 500549, 500999, 505004, 505049, 505499, 509999, 550004, 550049, 550499, 554999, 599999, 1000004, 1000049, 1000499, 1004999,
1049999, 1499999, 5000000, 5000009, 5000054, 5000099, 5000504, 5000549, 5000999, 5005004, 5005049, 5005499, 5009999, 5050004, 5050049, 5050499, 5054999, 5099999, 5500004, 5500049,5500499, 5504999,
5549999, 5999999, 10000004, 10000049, 10000499, 10004999, 10049999, 10499999, 14999999, 50000000, 50000009, 50000054, 50000099, 50000504, 50000549, 50000999, 50005004, 50005049, 50005499, 50009999,
50050004, 50050049, 50050499, 50054999, 50099999, 50500004, 50500049, 50500499, 50504999, 50549999, 50999999, 55000004, 55000049, 55000499, 55004999, 55049999, 55499999, ......}

{5, 14, 50, 59, 104, 149, 500, 509, 554, 599, 1004, 1049, 1499, 5000, 5009, 5054, 5099, 5504, 5549, 5999, 10004, 10049, 10499, 14999, 50000, 50009, 50054, 50099, 50504, 50549, 50999, 55004, 55049, 55499, 59999,
100004, 100049, 100499, 104999, 149999, 500000, 500009, 500054, 500099, 500504, 500549, 500999, 505004, 505049, 505499, 509999, 550004, 550049, 550499, 554999, 599999, 1000004, 1000049, 1000499, 1004999,
1049999, 1499999, 5000000, 5000009, 5000054, 5000099, 5000504, 5000549, 5000999, 5005004, 5005049, 5005499, 5009999, 5050004, 5050049, 5050499, 5054999, 5099999, 5500004, 5500049, 5500499, 5504999,
5549999, 5999999, 10000004, 10000049, 10000499, 10004999, 10049999, 10499999, 14999999, 50000000, 50000009, 50000054, 50000099, 50000504, 50000549, 50000999, 50005004, 50005049, 50005499, 50009999,
50050004, 50050049, 50050499, 50054999, 50099999, 50500004, 50500049, 50500499, 50504999, 50549999, 50999999, 55000004, 55000049, 55000499, 55004999, 55049999, 55499999, ......}

1. Select[Range[10^8], Total[IntegerDigits[2 # + 3]] == 4 &]

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王守恩

真题来了:  
A的数字和=11的倍数,  A+2的数字和还是=11的倍数。求助:  来几个(2个也行)。谢谢！
A=189999999999=99999999999 + 90000000000*1,
A=279999999999=99999999999 + 90000000000*2,  
A=369999999999=99999999999 + 90000000000*3,
A=459999999999=99999999999 + 90000000000*4,
A=549999999999=99999999999 + 90000000000*5,
A=639999999999=99999999999 + 90000000000*6,
A=729999999999=99999999999 + 90000000000*7,
A=819999999999=99999999999 + 90000000000*8,
A=909999999999=99999999999 + 90000000000*9,
A=1089999999999=99999999999 + 90000000000*11,
A=1179999999999=99999999999 + 90000000000*12,
A=1269999999999=99999999999 + 90000000000*13,
A=1359999999999=99999999999 + 90000000000*14,
A=1449999999999=99999999999 + 90000000000*15,
A=1539999999999=99999999999 + 90000000000*16,
A=1629999999999=99999999999 + 90000000000*17,
A=1719999999999=99999999999 + 90000000000*18,
A=1809999999999=99999999999 + 90000000000*19,
A=2079999999999=99999999999 + 90000000000*22,
A=2169999999999=99999999999 + 90000000000*23,
A=2259999999999=99999999999 + 90000000000*24,
{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99,
111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 125, 126, 127, 128, 129, 133, 134, 135, 136, 137, 138, 139, 144, 145, 146, 147, 148, 149, 155, 156, 157, 158, 159, 166, 167, 168,
169, 177, 178, 179, 188, 189, 199, 222, 223, 224, 225, 226, 227, 228, 229, 233, 234, 235, 236, 237, 238, 239, 244, 245, 246, 247, 248, 249, 255, 256, 257, 258, 259, 266, 267, 268, 269, 277,
278, 279, 288, 289, 299, 333, 334, 335, 336, 337, 338, 339, 344, 345, 346, 347, 348, 349, 355, 356, 357, 358, 359, 366, 367, 368, 369, 377, 378, 379, 388, 389, 399, 444, 445, 446, 447, 448,
449, 455, 456, 457, 458, 459, 466, 467, 468, 469, 477, 478, 479, 488, 489, 499, 555, 556, 557, 558, 559, 566, 567, 568, 569, 577, 578, 579, 588, 589, 599, 666, 667, 668, 669, 677, 678, 679,
688, 689, 699, 777, 778, 779, 788, 789, 799, 888, 889, 899, 999, 1111, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1122, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1133, 1134, 1135}

1. Select[Range[3000], Mod[Total[IntegerDigits[10^11 + 9*10^10 # - 1]], 11] ==  Mod[Total[IntegerDigits[10^11 + 9*10^10 # + 1]], 11] == 0 &]

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王守恩

wlc1 发表于 2024-8-17 18:56
A=2025

漂亮的数字串!   OEIS——没有这些漂亮的数字串。 OEIS——不应该有这些漂亮的数字串吗？

数字串(1)。1, 3, 5, 7, 9, 11, 33, 55, 77, 99, 111, 333, 555, 777, 999, 1111, 3333, 5555, 7777, 9999, 11111, 33333, 55555, 77777, 99999, 111111, 333333, 555555, 777777, 999999, 1111111, 3333333,
5555555, 7777777, 9999999, 11111111, 33333333, 55555555, 77777777, 99999999, 111111111, 333333333, 555555555, 777777777, 999999999, 1111111111, 3333333333, 5555555555, 7777777777, ...

数字串(2)。1, 4, 7, 11, 44, 77, 111, 444, 777, 1111, 4444, 7777, 11111, 44444, 77777, 111111, 444444, 777777, 1111111, 4444444, 7777777, 11111111, 44444444, 77777777, 111111111, 444444444,
777777777, 1111111111, 4444444444, 7777777777, 11111111111, 44444444444, 77777777777, 111111111111, 444444444444, 777777777777, 1111111111111, 4444444444444, 7777777777777, ...

数字串(3)。2, 5, 8, 22, 55, 88, 222, 555, 888, 2222, 5555, 8888, 22222, 55555, 88888, 222222, 555555, 888888, 2222222, 5555555, 8888888, 22222222, 55555555, 88888888, 222222222, 555555555,
888888888, 2222222222, 5555555555, 8888888888, 22222222222, 55555555555, 88888888888, 222222222222, 555555555555, 888888888888, 2222222222222, 5555555555555, 8888888888888, ...

数字串(4)。3, 6, 9, 33, 66, 99, 333, 666, 999, 3333, 6666, 9999, 33333, 66666, 99999, 333333, 666666, 999999, 3333333, 6666666, 9999999, 33333333, 66666666, 99999999, 333333333, 666666666,
999999999, 3333333333, 6666666666, 9999999999, 33333333333, 66666666666, 99999999999, 333333333333, 666666666666, 999999999999, 3333333333333, 6666666666666, 9999999999999, ...

数字串(5)。1, 6, 11, 66, 111, 666, 1111, 6666, 11111, 66666, 111111, 666666, 1111111, 6666666, 11111111, 66666666, 111111111, 666666666, 1111111111, 6666666666, 11111111111,
66666666666, 111111111111, 666666666666, 1111111111111, 6666666666666, 11111111111111, 66666666666666, 111111111111111, 666666666666666, 1111111111111111, 6666666666666666, ...

数字串(6)。2, 7, 22, 77, 222, 777, 2222, 7777, 22222, 77777, 222222, 777777, 2222222, 7777777, 22222222, 77777777, 222222222, 777777777, 2222222222, 7777777777, 22222222222,
77777777777, 222222222222, 777777777777, 2222222222222, 7777777777777, 22222222222222, 77777777777777, 222222222222222, 777777777777777, 2222222222222222, 7777777777777777, ...

数字串(7)。3, 8, 33, 88, 333, 888, 3333, 8888, 33333, 88888, 333333, 888888, 3333333, 8888888, 33333333, 88888888, 333333333, 888888888, 3333333333, 8888888888, 33333333333,
88888888888, 333333333333, 888888888888, 3333333333333, 8888888888888, 33333333333333, 88888888888888, 333333333333333, 888888888888888, 3333333333333333, 8888888888888888, ...

数字串(8)。4, 9, 44, 99, 444, 999, 4444, 9999, 44444, 99999, 444444, 999999, 4444444, 9999999, 44444444, 99999999, 444444444, 999999999, 4444444444, 9999999999, 44444444444,
99999999999, 444444444444, 999999999999, 4444444444444, 9999999999999, 44444444444444, 99999999999999, 444444444444444, 999999999999999, 4444444444444444, 9999999999999999, ...

数字串(9)。1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444,
55555, 66666, 77777, 88888, 99999, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 1111111, 2222222, 3333333, 4444444, 5555555, 6666666, 7777777, 8888888, 9999999, ...