数字串的通项公式

northwolves

王守恩 发表于 2024-8-2 15:29
a(1)=2——用1个1与1个2组成的2位置数有2个。
12,
21,

$C_{2n}^{n}=\{2,6,20,70,252,924,3432,12870,48620,184756,705432,2704156,10400600,40116600,155117520,601080390,2333606220,9075135300,35345263800,137846528820,538257874440,2104098963720,8233430727600,32247603683100,126410606437752,495918532948104,1946939425648112,7648690600760440,30067266499541040,118264581564861424\}$

王守恩

A088404——我来给它配个通项公式——还能快一点吗？谢谢！
{1, 10, 55, 100, 505, 550, 1000, 5005, 5050, 5500, 10000, 50005, 50050, 50500, 55000, 100000, 500005, 500050, 500500, 505000, 550000, 1000000, 5000005,
5000050, 5000500, 5005000, 5050000, 5500000, 10000000, 50000005, 50000050, 50000500, 50005000, 50050000, 50500000, 55000000, 100000000, ......}               
Select[Range[10^8], Total[IntegerDigits[2 # + 1]] == 3 &]

王守恩

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！

northwolves

王守恩 发表于 2024-8-18 11:04
A088404——我来给它配个通项公式——还能快一点吗？谢谢！
{1, 10, 55, 100, 505, 550, 1000, 5005, 5050, ...

1. Sort[r = 10^Range[0, 30]; Union[r, 5*Total /@ Subsets[r, {2}]]]

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northwolves

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

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northwolves

王守恩 发表于 2024-8-18 12:29
A+(A+1)的数字和=4。
A=6, 15, 51, 60, 105, 150, 501, 510, 555, 600, 1005, 1050, 1500, 5001, 5010, 505 ...

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

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{6,15,51,60,105,150,501,510,555,600,1005,1050,1500,5001,5010,5055,5100,5505,5550,6000,10005,10050,10500,15000,50001,50010,50055,50100,50505,50550,51000,55005,55050,55500,60000,100005,100050,100500,105000,150000,500001,500010,500055,500100,500505,500550,501000,505005,505050,505500,510000,550005,550050,550500,555000,600000,1000005,1000050,1000500,1005000,1050000,1500000,5000001,5000010,5000055,5000100,5000505,5000550,5001000,5005005,5005050,5005500,5010000,5050005,5050050,5050500,5055000,5100000,5500005,5500050,5500500,5505000,5550000,6000000,10000005,10000050,10000500,10005000,10050000,10500000,15000000,50000001,50000010,50000055,50000100,50000505,50000550,50001000,50005005,50005050,50005500,50010000,50050005,50050050,50050500,50055000,50100000,50500005,50500050,50500500,50505000,50550000,51000000,55000005,55000050,55000500,55005000,55050000,55500000,60000000,100000005,100000050,100000500,100005000,100050000,100500000,105000000,150000000,500000001,500000010,500000055,500000100,500000505,500000550,500001000,500005005,500005050,500005500,500010000,500050005,500050050,500050500,500055000,500100000,500500005,500500050,500500500,500505000,500550000,501000000,505000005,505000050,505000500,505005000,505050000,505500000,510000000,550000005,550000050,550000500,550005000,550050000,550500000,555000000,600000000,1000000005,1000000050,1000000500,1000005000,1000050000,1000500000,1005000000,1050000000,1500000000,5000000001,5000000010,5000000055,5000000100,5000000505,5000000550,5000001000,5000005005,5000005050,5000005500,5000010000,5000050005,5000050050,5000050500,5000055000,5000100000,5000500005,5000500050,5000500500,5000505000,5000550000,5001000000,5005000005,5005000050,5005000500,5005005000,5005050000,5005500000,5010000000,5050000005,5050000050,5050000500,5050005000,5050050000,5050500000,5055000000,5100000000,5500000005,5500000050,5500000500,5500005000,5500050000,5500500000,5505000000,5550000000,6000000000,10000000005,10000000050,10000000500,10000005000,10000050000,10000500000,10005000000,10050000000,10500000000,15000000000,50000000001,50000000010,50000000100,50000001000,50000010000,50000100000,50001000000,50010000000,50100000000,51000000000}

王守恩

northwolves 发表于 2024-8-18 22:03
{6,15,51,60,105,150,501,510,555,600,1005,1050,1500,5001,5010,5055,5100,5505,5550,6000,10005,1005 ...

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

三角数A=n(n+1)/2的数字和=3*1, A={2, 6, 15, 20, 24, 141, 200, 2000, 20000, 200000, 2000000,
三角数A=n(n+1)/2的数字和=3*2, A={3, 5, 14, 21, 66, 77, 201, 473, 2001, 15620, 20001, 200001, 2000001,
三角数A=n(n+1)/2的数字和=3*3, A={8, 9, 17, 18, 26, 35, 45, 53, 63, 80, 81, 89, 126, 144, 161, 162, 179, 206, 215, 224, 449, 458, 477, 666, 800, 801, 1421, 1575, 1620, 1673, 2006, 2015, 2195, 2835, 4473, 4733, 6326, 8000, 8001, 8126,  
三角数A=n(n+1)/2的数字和=3*4, A={11, 29, 33, 38, 42, 47, 51, 60, 65, 69, 78, 101, 110, 119, 146, 150, 155, 159, 164, 173, 195, 204, 245, 249, 258, 317, 326, 375, 402, 447, 510, 533, 600, 632, 663, 681, 722, 1001, 1095, 1122, 1266, 1415,
三角数A=n(n+1)/2的数字和=3*5, A={12, 23, 30, 32, 39, 41, 48, 50, 57, 59, 68, 75, 84, 86, 93, 95, 102, 104, 111, 113, 120, 122, 149, 156, 158, 167, 174, 185, 203, 210, 219, 221, 228, 237, 246, 255, 257, 266, 273, 284, 293, 300, 318, 320,
三角数A=n(n+1)/2的数字和=3*6, A={27, 36, 44, 54, 62, 71, 72, 90, 98, 99, 117, 134, 135, 143, 152, 153, 170, 171, 180, 197, 198, 207, 216, 225, 233, 242, 251, 260, 261, 269, 279, 287, 288, 297, 323, 324, 333, 341, 350, 359, 377, 378, 395,
三角数A=n(n+1)/2的数字和=3*7, A={56, 74, 83, 87, 92, 96, 105, 114, 123, 128, 137, 168, 177, 182, 186, 191, 209, 213, 218, 222, 240, 254, 263, 272, 276, 281, 285, 290, 294, 299, 303, 321, 330, 335, 339, 344, 348, 353, 357, 362, 366, 371,

把第一个数取出来(我的电脑只能出来这么几个)。OEIS没有这串数。
{2, 3, 8, 11, 12, 27, 56, 129, 107, 132, 309, 368, 627, 968, 1332, 3129, 3434, 5291, 8831, 13332, 18972, 28248, 37067, 77067, 107516, 140547, 278172, 368507,

王守恩

此序列共有 25332 个数。第一个数是 2。最后一个数是 987653201。OEIS——A046732。

1. Select[Prime[Range[600]], Length[Union[x = IntegerDigits[#]]] == Length[x] && PrimeQ[FromDigits[Reverse[x]]] &]

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{2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 107, 149, 157, 167, 179, 347, 359, 389, 701, 709, 739, 743, 751, 761, 769, 907, 937, 941, 953,
967, 971, 983, 1069, 1097, 1237, 1249, 1259, 1279, 1283, 1409, 1429, 1439, 1453, 1487, 1523, 1583, 1597, 1657, 1723, 1753, 1789, 1847,
1867, 1879, 3019, 3049, 3067, 3089, 3109, 3169, 3251, 3257, 3271, 3407, 3467, 3469, 3527, 3541, 3571, 3697, 3719, 3821, 3851, 3917, ...}

王守恩

用双面刻字(每面只能刻10个数码中的1个)的木块标记1到n，最少需要多少个木块？
1个木块可以标记1。(略去1个木块可以标记1,2)
2个木块可以标记1,2,3。(略去2个木块可以标记1,2,3,4)
3个木块可以标记1,2,3,4,5。
4个木块可以标记1,2,3,4,5,6,7。
5个木块可以标记1——9。
6个木块可以标记1——11。
7个木块可以标记1——33。
8个木块可以标记1——55。
9个木块可以标记1——77。

得到一串数。
1, 3, 5, 7, 9, 11, 33, 55, 77, 99, 111, 333, 555, 777, 999, 1111, ...