数字串的通项公式

王守恩

n = 1, 2, 3, 4, ...,  条件: Floor[Sqrt[n*a^2 + R(a)]] ≡ Floor[Sum[Sqrt[n + k], {k, a}]],  ≡ 表示有无穷个解。
R(a) 表示满足条件的最小解。 a = 1, 2, 3, 4, 5, 6, ......
a=1,R(1)=1。
a=2,R(2)=5。
a=3,R(3)=16。
a=4,R(4)=36。
a=5,R(5)=74。
a=6,R(6)=124。
a=7,R(7)=193。
a=8,R(8)=281。
a=9,R(9)=403。
......

得到一串数——1, 5, 16, 36, 74, 124, 193, 281, 403, 549, 724, 928, 1182, 1465, 1791, 2169, 2599, 3076, 3608, 4196, 4846, 5560, 6343, 7177, 8119, 9117, ......

Table[R = 1; While[Min[Table[Floor[Sqrt[n*a^2 + R]]/Floor[Sum[Sqrt[n + k], {k, a}]], {n, 800}]] ≠ 1, R++]; R, {a, 10}]

G[a_, M_] := Floor[Total[Sqrt[Range[a] + #]]] & /@ Range[M]; Q[a_, R_, d_] := AllTrue[Floor[Sqrt[Range@Length[d]*a^2 + R]] - d, # ≥ 0 &];
S[a_, M_, t_ : 400] := Module[{s = G[a, M], r = Floor[a^3/2]}, While[! Q[a, r, s], r++]; r]; Print[Table[S[a, 2000], {a, 20}]]

这代码可以改建吗？谢谢各位！！！

王守恩

8, 11, 14, 20, 26, 35, 50, 101, 203, 410, 1124, 3164, 12140, 50252, 266336, 2156330, 24220451, 412604363, ...... —— OEIS 没有这串数。
a(1)=8,
a(2)=11,——把8写成7进制数。
a(3)=14,——把11写成7进制数。
a(4)=20,——把14写成7进制数。
a(5)=26,——把20写成7进制数。
a(6)=35,
a(7)=50,

王守恩

接楼上——{8, 11, 14, 20, 26, 35, 50, 101, 203, 410, 1124, 3164, 12140, 50252, 266336, 2156330, 24220451, 412604363, 13140036113, 643423315121, 64325425640246, 16356234004310231, 13021120624163444663}
a(0)=8,
a(1)=11,——把8写成7进制数。
a(2)=14,——把11写成7进制数。
a(3)=20,——把14写成7进制数。
a(4)=26,——把20写成7进制数。
a(5)=35,
a(6)=50,
代码是这样——NestList[FromDigits[IntegerDigits[#, 7]] &, 8, 25]

王守恩

一般地。a(1)=把A进制 K 写成B进制。a(2)=把A进制a(1)写成B进制。a(3)=把A进制a(2)写成B进制。——要求:  K ≥ A,  B ＞ A。

代码(1)——NestList[FromDigits[IntegerDigits[#, B], A] &, K, n]——显示的是10进制数的a(n)。

代码(2)——IntegerLength[ NestList[FromDigits[IntegerDigits[#, B], A] &, K, n]]——显示的是10制数a(n)的位数。

譬如。a(1)=把w进制 w 写成(w+1)进制。a(2)=把w进制a(1)写成B(w+1)进制。a(3)=把w进制a(2)写成B(w+1)进制。

代码(1)——Table[NestList[FromDigits[IntegerDigits[#, w], w + 1] &, w, 14], {w, 3, 22}]
{3, 4, 5, 6, 8, 10, 17, 26, 42, 88, 265, 1061, 5386, 37973, 430226}, {4, 5, 6, 7, 8, 10, 12, 15, 18, 27, 38, 57, 86, 157, 291}, {5, 6, 7, 8, 9, 10, 12, 14, 16, 19, 22, 26, 37, 50, 72}, {06, 07, 08, 09, 10, 11, 12, 14, 16, 18, 21, 24, 28, 32, 37},
{07, 08, 09, 10, 11, 12, 13, 14, 16, 18, 20, 22, 25, 28, 32}, {08, 09, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 27, 30}, {09, 10, 11, 12, 13, 14, 15, 16,17, 18, 20, 22, 24, 26, 28}, {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28},
{11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28}, {12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28}, {13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28}, {14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28},
{15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, {17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, {18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32},
{19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, {20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, {21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, {22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}}

代码(2)——Table[IntegerLength[NestList[FromDigits[IntegerDigits[#, w], w + 1] &, w, 16]], {w, 3, 22}]
{1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 5, 6, 7, 9}, {1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4}, {1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3}, {1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2},
{1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2},
{2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2},
{2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2},
{2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}}