A086824 Least positive k such that k! >= n^k.

1, 1, 4, 7, 9, 12, 14, 17, 20, 22, 25, 28, 30, 33, 36, 38, 41, 44, 47, 49, 52, 55, 57, 60, 63, 65, 68, 71, 73, 76, 79, 82, 84, 87, 90, 92, 95, 98, 101, 103, 106, 109, 111, 114, 117, 119, 122, 125, 128, 130, 133, 136, 138, 141, 144, 147, 149, 152, 155, 157, 160, 163, 166

Offset: 0

Author

Benoit Cloitre, Aug 07 2003

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Variant of A065027. - R. J. Mathar, Sep 12 2008

Programs

Maple

a:= proc(n) option remember; local k; if n<0 then 1 else
      for k from a(n-1) while k! < n^k do od; k fi
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 15 2022

Mathematica

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

PARI

a(n)=if(n<2,1,k=1; while(k!
				

Formula

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

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

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

Extensions

Missing a(0)=1 inserted by Alois P. Heinz, Jan 15 2022