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Least positive k such that k! >= n^k.
2

%I #28 Mar 07 2024 08:43:20

%S 1,1,4,7,9,12,14,17,20,22,25,28,30,33,36,38,41,44,47,49,52,55,57,60,

%T 63,65,68,71,73,76,79,82,84,87,90,92,95,98,101,103,106,109,111,114,

%U 117,119,122,125,128,130,133,136,138,141,144,147,149,152,155,157,160,163,166

%N Least positive k such that k! >= n^k.

%C Suggested by Richard-Andre Jeannin (andre-jeannin.richard(AT)wanadoo.fr).

%H Alois P. Heinz, <a href="/A086824/b086824.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = e*n + O(log(n)); a(n+1)-a(n) = 2 or 3.

%F Conjecture: for n>3 a(n) = round(e*n-(1/2)*log(2*Pi*n)-1/n). - _Benoit Cloitre_, Dec 14 2005

%F Above conjecture is false: For n = 195 we have: a(n) = 526 < 527 = round(exp(1)*n -(1/2)*log(2*Pi*n)-1/n). - _Alois P. Heinz_, Jan 15 2022

%p a:= proc(n) option remember; local k; if n<0 then 1 else

%p for k from a(n-1) while k! < n^k do od; k fi

%p end:

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Jan 15 2022

%t f[n_] := Block[{k = 1}, While[k! < n^k, k++ ]; k]; Table[ f[n], {n, 62}] (* _Robert G. Wilson v_, Jun 12 2004 *)

%o (PARI) a(n)=if(n<2,1,k=1; while(k!<n^k,k++); k)

%Y Variant of A065027. - _R. J. Mathar_, Sep 12 2008

%K nonn

%O 0,3

%A _Benoit Cloitre_, Aug 07 2003

%E Missing a(0)=1 inserted by _Alois P. Heinz_, Jan 15 2022