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Least number k such that (10^n)^k < k!.
2

%I #4 Mar 30 2012 17:30:55

%S 2,25,269,2714,27177,271822,2718274,27182809,271828173,2718281817,

%T 27182818272,271828182832,2718281828444,27182818284575,

%U 271828182845887,2718281828459027,27182818284590433,271828182845904503

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

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

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

%t 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}]

%Y Cf. A065027, A011544.

%K nonn

%O 0,1

%A _Robert G. Wilson v_, Jul 13 2003