A113777 Minimal positive number m for which Sum_{k=1..m} k^n < m!.

3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90

Offset

0,1

Comments

a(n) > n, with a(n)/n -> 1 as n -> infinity. - Robert Israel, May 28 2018

Links

Robert Israel, Table of n, a(n) for n = 0..7485

Formula

Let S(n, m) = Sum_{k=1..m} k^n. Define a(n) = min{ m | S(n, m)<m! }.

Example

a(3)=6 because S(3,6)=441<720=6! but S(3,5)=225>=120=5! and so for S(3,j), j=0,1,2,3,4.

Maple

f:= proc(n) local k, L;
  L:= sum(k^n, k=1..n);
  for k from n+1 do
    L:= L + k^n;
    if L < k! then return k fi
  od
end proc:
map(f, [$0..100]); # Robert Israel, May 28 2018

Prog

(PARI) a(n) = {my(s = 0, ok = 0, m = 1); until (ok, s += m^n; if (s < m!, ok = 1, m++); ); return (m); } \\ Michel Marcus, Jul 15 2013

Crossrefs

Sequence in context: A288228 A002328 A039133 * A285659 A256421 A253200

Adjacent sequences: A113774 A113775 A113776 * A113778 A113779 A113780

Keyword

nonn

Author

Hieronymus Fischer, Jan 19 2006

Status

approved