一个三位数 [abc]，减去它的各位数码之和 a+b+c，得到一个三位数 [6x6]，求其中的 x

风花飘飘

哈哈哈，无限不循环就截断！

风花飘飘

王守恩 发表于 2026-9-25 11:29
OEIS有这串数——A282473——Multiples of 9 which cannot be expressed as the difference between a na ...

有通项式，一眼就看出来。

王守恩

风花飘飘 发表于 2026-9-28 12:16
有通项式，一眼就看出来。

A282473——9的倍数，不能表示为自然数k与其数字和s（k）之间的差。

90, 189, 288, 387, 486, 585, 684, 783, 882, 981, 990, 1089, 1188, 1287, 1386, 1485, 1584, 1683, 1782, 1881, 1980, 1989, 2088, 2187, 2286, 2385, 2484, 2583, 2682, 2781, 2880, 2979, 2988,

评论——根据实证观察，可以注意到连续项之间的差值通常是99。然而，当百位数为9时，这一差值就失效了，此时连续项之间的差值为9。不过，这一差值又会变回来。

链接——查尔斯·R·格雷特豪斯 IV，n表，a（n） 对 n = 1..10000Brilliant.org，这些数字真不错澳大利亚数学奥林匹克，2015年第2题，第84页。

示例——如果 k=90，则 k-s（k）=81。如果 k=100，则 k-s（k）=99。这是一个递增函数，因此 k-s（k）=90 是不可能实现的。

公式——okQ[n_] ：= Catch[ Do[ If[ x- Total@ IntegerDigits@ x == n， Throw@ False]， {x， n， n 9 IntegerLength[n]}];True];Select[9 Range[1000]， okQ[#] &] （* Giovanni Resta， 2017年2月27日 *）

作者——沙维尔·凯萨尔瓦尼，2017年2月16日

现状——批准了

风花飘飘

王守恩

下面是"通项公式"——好不容易搞了1个！可就是不满意！不好看！——各位！可愿意出手——改一改？谢谢！！

1. R = (10^# - 1)/9 &; maxBad = 100000000; maxS = Quotient[maxBad, 9] + 1; isBad[n_] := NoneTrue[Range[0, 9 (IntegerLength[n] + 1)], Total[IntegerDigits[n + #]] == # &];
   
2. bad = Select[9 (Union[Flatten[Table[11 a + 111 b + 1111 c + 11111 d + 111111 e + 1111111 f, {a, 0, 10}, {b, 0, 9}, {c, 0, 9}, {d, 0, 9}, {e,  0, 1}, {f, 0, 1}]],
   
3. Flatten[Table[10^m q + R[m] d - j, {m, 3, 7}, {q, 0, Ceiling[maxS/10^m] + 1}, {d, 1, 9}, {j, Range[m - 2]}]]] - 1), 0 < # <= maxBad && isBad[#] &];
   
4. indexedBad = Transpose[{Range@Length@bad, bad}];
   
5. Take[indexedBad, 10000]

复制代码

{{1, 90}, {2, 189}, {3, 288}, {4, 387}, {5, 486}, {6, 585}, {7, 684}, {8, 783}, {9, 882}, {10, 981}, {11, 990}, {12, 1089}, {13, 1188}, {14, 1287}, {15, 1386}, {16, 1485}, {17, 1584}, {18, 1683}, {19, 1782}, {20, 1881},
{21, 1980}, {22, 1989}, {23, 2088}, {24, 2187}, {25, 2286}, {26, 2385}, {27, 2484}, {28, 2583}, {29, 2682}, {30, 2781}, {31, 2880}, {32, 2979}, {33, 2988}, {34, 3087}, {35, 3186}, {36, 3285}, {37, 3384}, {38, 3483},
{39, 3582}, {40, 3681}, {41, 3780}, {42, 3879}, {43, 3978}, {44, 3987}, {45, 4086}, {46, 4185}, {47, 4284}, {48, 4383}, {49, 4482}, {50, 4581}, {51, 4680}, {52, 4779}, {53, 4878}, {54, 4977}, {55, 4986}, {56, 5085},
{57, 5184}, {58, 5283}, {59, 5382}, {60, 5481}, {61, 5580}, {62, 5679}, {63, 5778}, {64, 5877}, {65, 5976}, {66, 5985}, {67, 6084}, {68, 6183}, {69, 6282}, {70, 6381}, {71, 6480}, {72, 6579}, {73, 6678}, {74, 6777},
{75, 6876}, {76, 6975}, {77, 6984}, {78, 7083}, {79, 7182}, {80, 7281}, {81, 7380}, {82, 7479}, {83, 7578}, {84, 7677}, {85, 7776}, {86, 7875}, {87, 7974}, {88, 7983}, {89, 8082}, {90, 8181}, {91, 8280}, {92, 8379},
{93, 8478}, {94, 8577}, {95, 8676}, {96, 8775}, {97, 8874}, {98, 8973}, {99, 8982}, {100, 9081}, {101, 9180}, {102, 9279}, {103, 9378}, {104, 9477}, {105, 9576}, {106, 9675}, {107, 9774}, {108, 9873}, {109, 9972},
{110, 9981}, {111, 9990}, {112, 10089}, {113, 10188}, {114, 10287}, {115, 10386}, {116, 10485}, {117, 10584}, {118, 10683}, {119, 10782}, {120, 10881}, {121, 10980}, {122, 10989}, {123, 11088}, {124, 11187},
{125, 11286}, {126, 11385}, {127, 11484}, {128, 11583}, {129, 11682}, {130, 11781}, {131, 11880}, {132, 11979}, {133, 11988}, {134, 12087}, {135, 12186}, {136, 12285}, {137, 12384}, {138, 12483}, {139, 12582},
{140, 12681}, {141, 12780}, {142, 12879}, {143, 12978}, {144, 12987}, {145, 13086}, {146, 13185}, {147, 13284}, {148, 13383}, {149, 13482}, {150, 13581}, {151, 13680}, {152, 13779}, {153, 13878}, {154, 13977},
{155, 13986}, {156, 14085}, {157, 14184}, {158, 14283}, {159, 14382}, {160, 14481}, {161, 14580}, {162, 14679}, {163, 14778}, {164, 14877}, {165, 14976}, {166, 14985}, {167, 15084}, {168, 15183}, {169, 15282},
{170, 15381}, {171, 15480}, {172, 15579}, {173, 15678}, {174, 15777}, {175, 15876}, {176, 15975}, {177, 15984}, {178, 16083}, {179, 16182}, {180, 16281}, {181, 16380}, {182, 16479}, {183, 16578}, {184, 16677},
{185, 16776}, {186, 16875}, {187, 16974}, {188, 16983}, {189, 17082}, {190, 17181}, {191, 17280}, {192, 17379}, {193, 17478}, {194, 17577}, {195, 17676}, {196, 17775}, {197, 17874}, {198, 17973}, {199, 17982},
{200, 18081}, {201, 18180}, {202, 18279}, {203, 18378}, {204, 18477}, {205, 18576}, {206, 18675}, {207, 18774}, {208, 18873}, {209, 18972}, {210, 18981}, {211, 19080}, {212, 19179}, {213, 19278}, {214, 19377},
{215, 19476}, {216, 19575}, {217, 19674}, {218, 19773}, {219, 19872}, {220, 19971}, {221, 19980}, {222, 19989}, {223, 20088}, {224, 20187}, {225, 20286}, {226, 20385}, {227, 20484}, {228, 20583}, {229, 20682},
{230, 20781}, {231, 20880}, {232, 20979}, {233, 20988}, {234, 21087}, {235, 21186}, {236, 21285}, {237, 21384}, {238, 21483}, {239, 21582}, {240, 21681}, {241, 21780}, {242, 21879}, {243, 21978}, {244, 21987},
{245, 22086}, {246, 22185}, {247, 22284}, {248, 22383}, {249, 22482}, {250, 22581}, {251, 22680}, {252, 22779}, {253, 22878}, {254, 22977}, {255, 22986}, {256, 23085}, {257, 23184}, {258, 23283}, {259, 23382},
{260, 23481}, {261, 23580}, {262, 23679}, {263, 23778}, {264, 23877}, {265, 23976}, {266, 23985}, {267, 24084}, {268, 24183}, {269, 24282}, {270, 24381}, {271, 24480}, {272, 24579}, {273, 24678}, {274, 24777},
{275, 24876}, {276, 24975}, {277, 24984}, {278, 25083}, {279, 25182}, {280, 25281}, {281, 25380}, {282, 25479}, {283, 25578}, {284, 25677}, {285, 25776}, {286, 25875}, {287, 25974}, {288, 25983}, {289, 26082},
{290, 26181}, {291, 26280}, {292, 26379}, {293, 26478}, {294, 26577}, {295, 26676}, {296, 26775}, {297, 26874}, {298, 26973}, {299, 26982}, {300, 27081}, {301, 27180}, {302, 27279}, {303, 27378}, {304, 27477},
{305, 27576}, {306, 27675}, {307, 27774}, {308, 27873}, {309, 27972}, {310, 27981}, {311, 28080}, {312, 28179}, {313, 28278}, {314, 28377}, {315, 28476}, {316, 28575}, {317, 28674}, {318, 28773}, {319, 28872},
{320, 28971}, {321, 28980}, {322, 29079}, {323, 29178}, {324, 29277}, {325, 29376}, {326, 29475}, {327, 29574}, {328, 29673}, {329, 29772}, {330, 29871}, {331, 29970}, {332, 29979}, {333, 29988}, {334, 30087},
{335, 30186}, {336, 30285}, {337, 30384}, {338, 30483}, {339, 30582}, {340, 30681}, {341, 30780}, {342, 30879}, {343, 30978}, {344, 30987}, {345, 31086}, {346, 31185}, {347, 31284}, {348, 31383}, {349, 31482},
{350, 31581}, {351, 31680}, {352, 31779}, {353, 31878}, {354, 31977}, {355, 31986}, {356, 32085}, {357, 32184}, {358, 32283}, {359, 32382}, {360, 32481}, {361, 32580}, {362, 32679}, {363, 32778}, {364, 32877},
{365, 32976}, {366, 32985}, {367, 33084}, {368, 33183}, {369, 33282}, {370, 33381}, {371, 33480}, {372, 33579}, {373, 33678}, {374, 33777}, {375, 33876}, {376, 33975}, {377, 33984}, {378, 34083}, {379, 34182},
{380, 34281}, {381, 34380}, {382, 34479}, {383, 34578}, {384, 34677}, {385, 34776}, {386, 34875}, {387, 34974}, {388, 34983}, {389, 35082}, {390, 35181}, {391, 35280}, {392, 35379}, {393, 35478}, {394, 35577},
{395, 35676}, {396, 35775}, {397, 35874}, {398, 35973}, {399, 35982}, {400, 36081}, {401, 36180}, {402, 36279}, {403, 36378}, {404, 36477}, {405, 36576}, {406, 36675}, {407, 36774}, {408, 36873}, {409, 36972},
{410, 36981}, {411, 37080}, {412, 37179}, {413, 37278}, {414, 37377}, {415, 37476}, {416, 37575}, {417, 37674}, {418, 37773}, {419, 37872}, {420, 37971}, {421, 37980}, {422, 38079}, {423, 38178}, {424, 38277},
{425, 38376}, {426, 38475}, {427, 38574}, {428, 38673}, {429, 38772}, {430, 38871}, {431, 38970}, {432, 38979}, {433, 39078}, {434, 39177}, {435, 39276}, {436, 39375}, {437, 39474}, {438, 39573}, {439, 39672},
{440, 39771}, {441, 39870}, {442, 39969}, {443, 39978}, {444, 39987}, {445, 40086}, {446, 40185}, {447, 40284}, {448, 40383}, {449, 40482}, {450, 40581}, {451, 40680}, {452, 40779}, {453, 40878}, {454, 40977},
{455, 40986}, {456, 41085}, {457, 41184}, {458, 41283}, {459, 41382}, {460, 41481}, {461, 41580}, {462, 41679}, {463, 41778}, {464, 41877}, {465, 41976}, {466, 41985}, {467, 42084}, {468, 42183}, {469, 42282},
{470, 42381}, {471, 42480}, {472, 42579}, {473, 42678}, {474, 42777}, {475, 42876}, {476, 42975}, {477, 42984}, {478, 43083}, {479, 43182}, {480, 43281}, {481, 43380}, {482, 43479}, {483, 43578}, {484, 43677},
{485, 43776}, {486, 43875}, {487, 43974}, {488, 43983}, {489, 44082}, {490, 44181}, {491, 44280}, {492, 44379}, {493, 44478}, {494, 44577}, {495, 44676}, {496, 44775}, {497, 44874}, {498, 44973}, {499, 44982},
{500, 45081}, {501, 45180}, {502, 45279}, {503, 45378}, {504, 45477}, {505, 45576}, {506, 45675}, {507, 45774}, {508, 45873}, {509, 45972}, {510, 45981}, {511, 46080}, {512, 46179}, {513, 46278}, {514, 46377},
{515, 46476}, {516, 46575}, {517, 46674}, {518, 46773}, {519, 46872}, {520, 46971}, {521, 46980}, {522, 47079}, {523, 47178}, {524, 47277}, {525, 47376}, {526, 47475}, {527, 47574}, {528, 47673}, {529, 47772},
{
{700, 63081}, {701, 63180}, {702, 63279}, {703, 63378}, {704, 63477}, {705, 63576}, {706, 63675}, {707, 63774}, {708, 63873}, {709, 63972}, {710, 63981}, {711, 64080}, {712, 64179}, {713, 64278}, {714, 64377},
{715, 64476}, {716, 64575}, {717, 64674}, {718, 64773}, {719, 64872}, {720, 64971}, {721, 64980}, {722, 65079}, {723, 65178}, {724, 65277}, {725, 65376}, {726, 65475}, {727, 65574}, {728, 65673}, {729, 65772},
{730, 65871}, {731, 65970}, {732, 65979}, {733, 66078}, {734, 66177}, {735, 66276}, {736, 66375}, {737, 66474}, {738, 66573}, {739, 66672}, {740, 66771}, {741, 66870}, {742, 66969}, {743, 66978}, {744, 67077},
{745, 67176}, {746, 67275}, {747, 67374}, {748, 67473}, {749, 67572}, {750, 67671}, {751, 67770}, {752, 67869}, {753, 67968}, {754, 67977}, {755, 68076}, {756, 68175}, {757, 68274}, {758, 68373}, {759, 68472},
{760, 68571}, {761, 68670}, {762, 68769}, {763, 68868}, {764, 68967}, {765, 68976}, {766, 69075}, {767, 69174}, {768, 69273}, {769, 69372}, {770, 69471}, {771, 69570}, {772, 69669}, {773, 69768}, {774, 69867},
{775, 69966}, {776, 69975}, {777, 69984}, {778, 70083}, {779, 70182}, {780, 70281}, {781, 70380}, {782, 70479}, {783, 70578}, {784, 70677}, {785, 70776}, {786, 70875}, {787, 70974}, {788, 70983}, {789, 71082},
{790, 71181}, {791, 71280}, {792, 71379}, {793, 71478}, {794, 71577}, {795, 71676}, {796, 71775}, {797, 71874}, {798, 71973}, {799, 71982}, {800, 72081}, {801, 72180}, {802, 72279}, {803, 72378}, {804, 72477},
{805, 72576}, {806, 72675}, {807, 72774}, {808, 72873}, {809, 72972}, {810, 72981}, {811, 73080}, {812, 73179}, {813, 73278}, {814, 73377}, {815, 73476}, {816, 73575}, {817, 73674}, {818, 73773}, {819, 73872},
{820, 73971}, {821, 73980}, {822, 74079}, {823, 74178}, {824, 74277}, {825, 74376}, {826, 74475}, {827, 74574}, {828, 74673}, {829, 74772}, {830, 74871}, {831, 74970}, {832, 74979}, {833, 75078}, {834, 75177},
{835, 75276}, {836, 75375}, {837, 75474}, {838, 75573}, {839, 75672}, {840, 75771}, {841, 75870}, {842, 75969}, {843, 75978}, {844, 76077}, {845, 76176}, {846, 76275}, {847, 76374}, {848, 76473}, {849, 76572},
{850, 76671}, {851, 76770}, {852, 76869}, {853, 76968}, {854, 76977}, {855, 77076}, {856, 77175}, {857, 77274}, {858, 77373}, {859, 77472}, {860, 77571}, {861, 77670}, {862, 77769}, {863, 77868}, {864, 77967},
{865, 77976}, {866, 78075}, {867, 78174}, {868, 78273}, {869, 78372}, {870, 78471}, {871, 78570}, {872, 78669}, {873, 78768}, {874, 78867}, {875, 78966}, {876, 78975}, {877, 79074}, {878, 79173}, {879, 79272},
{880, 79371}, {881, 79470}, {882, 79569}, {883, 79668}, {884, 79767}, {885, 79866}, {886, 79965}, {887, 79974}, {888, 79983}, {889, 80082}, {890, 80181}, {891, 80280}, {892, 80379}, {893, 80478}, {894, 80577},
{895, 80676}, {896, 80775}, {897, 80874}, {898, 80973}, {899, 80982}, {900, 81081}, {901, 81180}, {902, 81279}, {903, 81378}, {904, 81477}, {905, 81576}, {906, 81675}, {907, 81774}, {908, 81873}, {909, 81972},
{910, 81981}, {911, 82080}, {912, 82179}, {913, 82278}, {914, 82377}, {915, 82476}, {916, 82575}, {917, 82674}, {918, 82773}, {919, 82872}, {920, 82971}, {921, 82980}, {922, 83079}, {923, 83178}, {924, 83277},
{925, 83376}, {926, 83475}, {927, 83574}, {928, 83673}, {929, 83772}, {930, 83871}, {931, 83970}, {932, 83979}, {933, 84078}, {934, 84177}, {935, 84276}, {936, 84375}, {937, 84474}, {938, 84573}, {939, 84672},
{940, 84771}, {941, 84870}, {942, 84969}, {943, 84978}, {944, 85077}, {945, 85176}, {946, 85275}, {947, 85374}, {948, 85473}, {949, 85572}, {950, 85671}, {951, 85770}, {952, 85869}, {953, 85968}, {954, 85977},
{955, 86076}, {956, 86175}, {957, 86274}, {958, 86373}, {959, 86472}, {960, 86571}, {961, 86670}, {962, 86769}, {963, 86868}, {964, 86967}, {965, 86976}, {966, 87075}, {967, 87174}, {968, 87273}, {969, 87372},
{970, 87471}, {971, 87570}, {972, 87669}, {973, 87768}, {974, 87867}, {975, 87966}, {976, 87975}, {977, 88074}, {978, 88173}, {979, 88272}, {980, 88371}, {981, 88470}, {982, 88569}, {983, 88668}, {984, 88767},
{985, 88866}, {986, 88965}, {987, 88974}, {988, 89073}, {989, 89172}, {990, 89271}, {991, 89370}, {992, 89469}, {993, 89568}, {994, 89667}, {995, 89766}, {996, 89865}, {997, 89964}, {998, 89973}, {999, 89982},

1. Transpose[{Range@Length@#, #}] &[Complement[Range[9, 900081, 9], Array[# - Tr@IntegerDigits@# &, 900081 + 9 IntegerLength@900081]]]

复制代码

——这个不是"通项公式"——可就是好看！——OEIS没有"通项公式"。

王守恩

大胆往前走！走！！走！！！——OEIS可没有这样惬意的想法！！
{{1, 90},
{11, 990},
{111, 9990},
{1111, 99990},
{11111, 999990},
{111111, 9999990},
{1111111, 99999990},
{11111111, 999999990},
{111111111, 9999999990},
{1111111111, 99999999990},
{11111111111, 999999999990},
{111111111111, 9999999999990},
{1111111111111, 99999999999990},
{11111111111111, 999999999999990},
{111111111111111, 9999999999999990},
{1111111111111111, 99999999999999990},
{11111111111111111, 999999999999999990},
{111111111111111111, 9999999999999999990},
{1111111111111111111, 99999999999999999990},
{11111111111111111111, 999999999999999999990},
{111111111111111111111, 9999999999999999999990},
{1111111111111111111111, 99999999999999999999990},
{11111111111111111111111, 999999999999999999999990},
{111111111111111111111111, 9999999999999999999999990},
{1111111111111111111111111, 99999999999999999999999990},
{11111111111111111111111111, 999999999999999999999999990},
{111111111111111111111111111, 9999999999999999999999999990},
{1111111111111111111111111111, 99999999999999999999999999990},
{11111111111111111111111111111, 999999999999999999999999999990}}
Table[{(10^k - 1)/9, 10 (10^k - 1)}, {k, 29}]

王守恩

说走咱就走哇！！！你有我有全都有哇！！！
{{9, 882},
{99, 8982},
{999, 89982},
{9999, 899982},
{99999, 8999982},
{999999, 89999982},
{9999999, 899999982},
{99999999, 8999999982},
{999999999, 89999999982},
{9999999999, 899999999982},
{99999999999, 8999999999982},
{999999999999, 89999999999982},
{9999999999999, 899999999999982},
{99999999999999, 8999999999999982},
{999999999999999, 89999999999999982},
{9999999999999999, 899999999999999982},
{99999999999999999, 8999999999999999982},
{999999999999999999, 89999999999999999982}}
Table[{10^k - 1, 90*10^k - 18}, {k, 18}]

王守恩

出道难题——后面的怎么填？
{3, 288},
{23, 2088},
{323, 29178},
{2323, 209178},
{32323, 2909268},
{232323, 20909268},
{3232323, 290909358},
{23232323, 2090909358},
{323232323, 29090909448},
{2323232323, 209090909448},
{32323232323, 2909090909538},
{232323232323, 20909090909538},
{3232323232323, 290909090909628},
{23232323232323, 2090909090909628},
{323232323232323,

王守恩

我们称这串数{90, 189, 288, 387, 486, 585, 684, 783, 882, 981, 990, 1089, 1188, 1287, 1386, 1485, 1584, 1683, ......为潇洒数。

十位数字 = 9 最小潇洒数 = 90。
十位数字 = 8 最小潇洒数 = 189。
十位数字 = 7 最小潇洒数 = 2979。
十位数字 = 6 最小潇洒数 = 39969。
十位数字 = 5 最小潇洒数 = 499959。
十位数字 = 4 最小潇洒数 = 5999949。
十位数字 = 3 最小潇洒数 = 69999939。
十位数字 = 2 最小潇洒数 = 799999929。
十位数字 = 1 最小潇洒数 = 8999999919。
十位数字 = 0 最小潇洒数 = 99999999909。

请你补个证明。

王守恩

{{2, 189},
{12, 1089},
{212, 19179},
{1212, 109179},
{21212, 1909269},
{121212, 10909269},
{2121212, 190909359},
{12121212, 1090909359},
{212121212, 19090909449},
{1212121212, 109090909449},
{21212121212, 1909090909539},
{121212121212, 10909090909539},
{2121212121212, 190909090909629},
{12121212121212, 1090909090909629},
{212121212121212, 19090909090909719},
{1212121212121212, 109090909090909719},
{21212121212121212, 1909090909090909809},
{121212121212121212, 10909090909090909809},
{2121212121212121212, 190909090909090909899},
{12121212121212121212, 1090909090909090909899},
{212121212121212121212, 19090909090909090909989},
{1212121212121212121212, 109090909090909090909989},
{21212121212121212121212, 1909090909090909090910079},
{121212121212121212121212, 10909090909090909090910079},
{2121212121212121212121212, 190909090909090909090910169},
{12121212121212121212121212, 1090909090909090909090910169}}