A085830 Least number k such that (10^n)^k < k!.

2, 25, 269, 2714, 27177, 271822, 2718274, 27182809, 271828173, 2718281817, 27182818272, 271828182832, 2718281828444, 27182818284575, 271828182845887, 2718281828459027, 27182818284590433, 271828182845904503

Offset: 0

Author

Robert G. Wilson v, Jul 13 2003

Keywords

• nonn

Comments

A085830(n) = A065027(10^n). This should confirm that the lim n -> infinity of A065027(n)/n -> e from below.

a(63) differs from the Floor(10^63* e) by only 33.

Crossrefs

Cf. A065027, A011544.

Programs

Mathematica

LogBaseBStirling[b_, n_] := Block[{}, N[ Log[b, 2*Pi*n]/2 + n*Log[b, n/E] + Log[b, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5)], 64]]; f[0] = 2; f[n_] := f[n] = Block[{k = 10*g[n - 1]}, While[ LogBaseBStirling[10^n, k] <= k, k++ ]; k]; Table[ f[n], {n, 1, 18}]