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数学研发论坛»论坛 › 【数学研究】 › 小题大作 › 数字串的通项公式
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楼主: 王守恩

[原创] 数字串的通项公式

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王守恩
 楼主| 发表于 2026-1-5 15:46:26 | 显示全部楼层
Table[FindInstance[{a + 2 b == n^2, n - 2 <= a <= b}, {b, a}, Integers, 1], {n, 20}]
{{{b -> 1, a -> -1}}, {{b -> 2, a -> 0}}, {{b -> 3,  a -> 3}}, {{b -> 6, a -> 4}}, {{b -> 9, a -> 7}},
{{b -> 12, a -> 12}}, {{b -> 17, a -> 15}}, {{b -> 22, a -> 20}}, {{b -> 27, a -> 27}}, {{b -> 34, a -> 32}},
{{b -> 41, a -> 39}}, {{b -> 48, a -> 48}}, {{b -> 57, a -> 55}}, {{b -> 66, a -> 64}}, {{b -> 75, a -> 75}},
{{b -> 86, a -> 84}}, {{b -> 97, a -> 95}}, {{b -> 108, a -> 108}}, {{b -> 121, a -> 119}}, {{b -> 134, a -> 132}}}

b={1, 2, 3, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 121, 134, 147, 162, 177, 192, 209, 226, 243, 262, 281, 300, 321, 342, 363, 386, 409, 432, 457, 482, 507, 534, 561, 588, 617, 646, 675, 706, 737, 768, 801, 834, 867}
Table[Ceiling[n^2/3], {n, 60}]

a={-1, 0, 3, 4, 7, 12, 15, 20, 27, 32, 39, 48, 55, 64, 75, 84, 95, 108, \119, 132, 147, 160, 175, 192, 207, 224, 243, 260, 279, 300, 319, 340, 363, 384, 407, 432, 455, 480, 507, 532, 559, 588, 615, 644, 675, 704, 735, 768, 799, 832, 867}
Table[Ceiling[n^2/3] - 2 Ceiling[n/3] + 2 Floor[n/3], {n, 60}]
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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王守恩
 楼主| 发表于 4 天前 | 显示全部楼层
\(00=1^3-1^3-0^3-0^3+0^3\)
\(01=1^3-1^3-0^3-0^3+1^3\)
\(02=0^3-2^3+1^3+1^3+2^3\)
\(03=3^3-5^3+4^3+4^3-3^3\)
\(04=3^3+1^3-2^3-2^3-2^3\)
\(05=2^3+0^3-1^3-1^3-1^3\)
\(06=2^3+0^3-1^3-1^3+0^3\)
\(07=2^3+0^3-1^3-1^3+1^3\)
\(08=1^3-1^3-0^3-0^3+2^3\)
\(09=3^3-4^3+3^3+3^3-2^3\)
\(10=4^3+2^3-3^3-3^3-2^3\)
\(11=3^3+1^3-2^3-2^3-1^3\)
\(12=3^3+1^3-2^3-2^3+0^3\)
\(13=3^3+1^3-2^3-2^3+1^3\)
\(14=2^3+0^3-1^3-1^3+2^3\)
\(15=3^3-3^3+2^3+2^3-1^3\)
\(16=5^3+3^3-4^3-4^3-2^3\)
\(17=4^3+2^3-3^3-3^3-1^3\)
\(18=4^3+2^3-3^3-3^3-0^3\)
\(19=4^3+2^3-3^3-3^3+1^3\)
\(20=3^3+1^3-2^3-2^3+2^3\)
\(21=0^3-2^3+1^3+1^3+3^3\)
\(22=6^3+4^3-5^3-5^3-2^3\)
\(23=5^3+3^3-4^3-4^3-1^3\)
\(24=5^3+3^3-4^3-4^3-0^3\)
\(25=5^3+3^3-4^3-4^3+1^3\)
\(26=4^3+2^3-3^3-3^3+2^3\)
\(27=1^3-1^3-0^3-0^3+3^3\)
\(28=7^3+5^3-6^3-6^3-2^3\)
\(29=6^3+4^3-5^3-5^3-1^3\)
\(30=6^3+4^3-5^3-5^3-0^3\)
\(31=6^3+4^3-5^3-5^3+1^3\)
\(32=5^3+3^3-4^3-4^3+2^3\)
\(33=2^3+0^3-1^3-1^3+3^3\)
\(34=8^3+6^3-7^3-7^3-2^3\)
\(35=7^3+5^3-6^3-6^3-1^3\)
\(36=7^3+5^3-6^3-6^3-0^3\)
\(37=7^3+5^3-6^3-6^3+1^3\)
\(38=6^3+4^3-5^3-5^3+2^3\)
\(39=3^3+1^3-2^3-2^3+3^3\)
\(40=9^3+7^3-8^3-8^3-2^3\)
\(41=8^3+6^3-7^3-7^3-1^3\)
\(42=8^3+6^3-7^3-7^3-0^3\)
\(43=8^3+6^3-7^3-7^3+1^3\)
\(44=7^3+5^3-6^3-6^3+2^3\)
\(45=4^3+2^3-3^3-3^3+3^3\)

\(n=\bigg(\big\lfloor\frac{n}{6}\big\rfloor + 1 - x\bigg)^3 +\bigg (\big\lfloor\frac{n}{6}\big\rfloor - 1 -  x\bigg)^3+ \bigg(-\big\lfloor\frac{n}{6}\big\rfloor + x\bigg)^3 + \bigg(-\big\lfloor\frac{n}{6}\big\rfloor + x\bigg)^3 + \bigg(Mod[n, 6] - 6\big\lfloor\frac{Mod[n, 6]}{4}\big\rfloor\bigg)^3\)

\(x=\big\lfloor\frac{Mod[n, 6]}{2}\big\rfloor + 3\big\lfloor\frac{Mod[n, 6]}{3}\big\rfloor - 7\big\lfloor\frac{Mod[n, 6]}{4}\big\rfloor +\big \lfloor\frac{Mod[n, 6]}{5}\big\rfloor\)

只要一串数就够了——{-1, -1, -2, -5, 1, 0, 0, 0, -1, -4, 2, 1, 1, 1, 0, -3, 3, 2, 2, 2, 1, -2, 4, 3, 3, 3, 2, -1, 5, 4, 4, 4, 3, 0, 6, 5, 5, 5, 4, 1, 7, 6, 6, 6, 5, 2, 8, 7, 7, 7, 6, 3, 9, 8, 8, 8, 7, 4, 10, 9, 9, 9, 8, 5, 11, 10, 10, 10, 9, 6, 12, 11, 11, 11, 10, 7, 13}

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {-1, -1, -2, -5, 1, 0, 0}, 90]
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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王守恩
 楼主| 发表于 昨天 06:38 | 显示全部楼层
求助！！！我们能把第5个数的1去掉吗？？？

\(00=1^3-1^3-0^3-0^3-0^3\)
\(01=1^3-1^3-0^3-0^3+1^3\)
\(02=0^3-5^3+4^3+4^3-1^3\)
\(03=0^3-5^3+4^3+4^3-0^3\)
\(04=0^3-5^3+4^3+4^3+1^3\)
\(05=2^3+0^3-1^3-1^3-1^3\)
\(06=2^3+0^3-1^3-1^3-0^3\)
\(07=2^3+0^3-1^3-1^3+1^3\)
\(08=1^3-3^3+3^3+2^3-1^3\)
\(09=1^3-3^3+3^3+2^3-0^3\)
\(10=1^3-3^3+3^3+2^3+1^3\)
\(11=3^3+1^3-2^3-2^3-1^3\)
\(12=3^3+1^3-2^3-2^3-0^3\)
\(13=3^3+1^3-2^3-2^3+1^3\)
\(14=2^3-1^3+2^3-0^3-1^3\)
\(15=2^3-1^3+2^3-0^3-0^3\)
\(16=2^3-1^3+2^3-0^3+1^3\)
\(17=4^3+2^3-3^3-3^3-1^3\)
\(18=4^3+2^3-3^3-3^3-0^3\)
\(19=4^3+2^3-3^3-3^3+1^3\)
\(20=3^3+1^3+1^3-2^3-1^3\)
\(21=3^3+1^3+1^3-2^3-0^3\)
\(22=3^3+1^3+1^3-2^3+1^3\)
\(23=5^3+3^3-4^3-4^3-1^3\)
\(24=5^3+3^3-4^3-4^3-0^3\)
\(25=5^3+3^3-4^3-4^3+1^3\)
\(26=4^3+3^3-0^3-4^3-1^3\)
\(27=4^3+3^3-0^3-4^3-0^3\)
\(28=4^3+3^3-0^3-4^3+1^3\)
\(29=6^3+4^3-5^3-5^3-1^3\)
\(30=6^3+4^3-5^3-5^3-0^3\)
\(31=6^3+4^3-5^3-5^3+1^3\)
\(32=5^3+5^3-1^3-6^3-1^3\)
\(33=5^3+5^3-1^3-6^3-0^3\)
\(34=5^3+5^3-1^3-6^3-1^3\)
\(35=7^3+5^3-6^3-6^3-1^3\)
\(36=7^3+5^3-6^3-6^3-0^3\)
\(37=7^3+5^3-6^3-6^3+1^3\)
\(38=6^3+7^3-2^3-8^3-1^3\)
\(39=6^3+7^3-2^3-8^3-0^3\)
\(40=6^3+7^3-2^3-8^3+1^3\)
\(41=8^3+6^3-7^3-7^3-1^3\)
\(42=8^3+6^3-7^3-7^3-0^3\)
\(43=8^3+6^3-7^3-7^3+1^3\)

Table[With[{m = Quotient[n + 1, 6], r = Mod[n, 6]}, If[r < 2 || r == 5, {n, m + 1, m - 1, m, m, If[r == 5, -1, r]}, {n, m, 2 m - 5, m - 4, 2 m - 4, r - 3}]], {n, 0, 100}]
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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王守恩
 楼主| 发表于 15 小时前 | 显示全部楼层
A344308 Numbers k such that A205791(k) = k+1.
{1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 23, 26, 29, 30, 34, 35, 37, 38, 39, 42, 43, 46, 47, 51, 53, 57, 58, 59, 65, 67, 69, 70, 73, 74, 78, 79, 83, 85, 86, 87, 89, 91, 94, 95, 97, 102, 103, 105, 106, 107, 109, 111, 113, 114, 115, 118, 119, 127, 129,
130, 133, 134, 137, 138, 139, 141, 145, 146, 149, 157, 158, 159, 161, 163, 166, 167, 170, 173, 174, 177, 178, 179, 182, 185, 190, 193, 194, 195, 197, 199, 201, 203, 206, 210, 214, 215, 218, 219, 221, 222, 223, 226, 227, 229, 230, 233, 235, 237,
238, 239, 247, 249, 254, 255, 257, 258, 259, 263, 265, 266, 267, 269, 273, 274, 277, 278, 282, 283, 285, 290, 291, 293, 295, 298, 299, 301, 307, 309, 313, 314, 317, 318, 321, 322, 323, 326, 327, 329, 334, 335, 337, 339, 345, 346, 347, 349, ...}
Select[Range@200, (k=1; While[FreeQ[Mod[Table[k^5-j^5, {j, k-1}], #], 0], k++]; k)==#+1&]—— (* Giorgos Kalogeropoulos, May 14 2021 *)
                    Array[(k=1; While[FreeQ[Mod[Table[k^5-j^5, {j, k-1}], #], 0], k++]; k)&, 100]—— (* Giorgos Kalogeropoulos, May 14 2021 *)

相同的数字串——Q[n_] := Sort[PowerMod[#, 5, n] & /@ Range@n] == Range@n - 1; Select[Range@360, Q]——这个算法快一些。
{1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 23, 26, 29, 30, 34, 35, 37, 38, 39, 42, 43, 46, 47, 51, 53, 57, 58, 59, 65, 67, 69, 70, 73, 74, 78, 79, 83, 85, 86, 87, 89, 91, 94, 95, 97, 102, 103, 105, 106, 107, 109, 111, 113, 114, 115, 118, 119, 127, 129,
130, 133, 134, 137, 138, 139, 141, 145, 146, 149, 157, 158, 159, 161, 163, 166, 167, 170, 173, 174, 177, 178, 179, 182, 185, 190, 193, 194, 195, 197, 199, 201, 203, 206, 210, 214, 215, 218, 219, 221, 222, 223, 226, 227, 229, 230, 233, 235, 237,
238, 239, 247, 249, 254, 255, 257, 258, 259, 263, 265, 266, 267, 269, 273, 274, 277, 278, 282, 283, 285, 290, 291, 293, 295, 298, 299, 301, 307, 309, 313, 314, 317, 318, 321, 322, 323, 326, 327, 329, 334, 335, 337, 339, 345, 346, 347, 349, ...}

不同的数字串——Q[n_] := Sort[PowerMod[#, 7, n] & /@ Range@n] == Range@n - 1; Select[Range@360, Q]——这个算法快多了——OEIS没有这串数。
{1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 46, 47, 51, 53, 55, 57, 59, 61, 62, 65, 66, 67, 69, 70, 73, 74, 77, 78, 79, 82, 83, 85, 89, 91, 93, 94, 95, 97, 101, 102, 103,
105, 106, 107, 109, 110, 111, 114, 115, 118, 119, 122, 123, 130, 131, 133, 134, 137, 138, 139, 141, 143, 146, 149, 151, 154, 155, 157, 158, 159, 161, 163, 165, 166, 167, 170, 173, 177, 178, 179, 181, 182, 183, 185, 186,
187, 190, 191, 193, 194, 195, 199, 201, 202, 205, 206, 209, 210, 214, 217, 218, 219, 221, 222, 223, 227, 229, 230, 231, 233, 235, 237, 238, 241, 246, 247, 249, 251, 253, 255, 257, 259, 262, 263, 265, 266, 267, 269, 271, 273,
274, 277, 278, 282, 283, 285, 286, 287, 291, 293, 295, 298, 299, 302, 303, 305, 307, 309, 310, 311, 313, 314, 317, 318, 321, 322, 323, 326, 327, 329, 330, 331, 334, 335, 341, 345, 346, 347, 349, 353, 354, 357, 358, 359, ...}

不同的数字串——A[n_] := Sort[PowerMod[#, 9, n] & /@ Range@n] == Range@n - 1; Select[Range@360, A]——B[n_] := Sort[PowerMod[#, 3, n] & /@ Range@n] == Range@n - 1; Select[Range@360, B]——答案是同一串数。
{1, 2, 3, 5, 6, 10, 11, 15, 17, 22, 23, 29, 30, 33, 34, 41, 46, 47, 51, 53, 55, 58, 59, 66, 69, 71, 82, 83, 85, 87, 89, 94, 101, 102, 106, 107, 110, 113, 115, 118, 123, 131, 137, 138, 141, 142, 145, 149, 159, 165, 166, 167, 170, 173, 174, 177, 178, 179,
187, 191, 197, 202, 205, 213, 214, 226, 227, 230, 233, 235, 239, 246, 249, 251, 253, 255, 257, 262, 263, 265, 267, 269, 274, 281, 282, 290, 293, 295, 298, 303, 311, 317, 318, 319, 321, 330, 334, 339, 345, 346, 347, 353, 354, 355, 358, 359, ...}

Q[n_] := Sort[PowerMod[#, 2, n] & /@ Range@n] == Range@n - 1; Select[Range@360, Q]
{1, 2}——这个更伟大了——只有{1, 2}两个数。—— "2" 改 "任意偶数" ,  还是{1, 2}两个数。
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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