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%I #14 May 28 2018 03:25:27
%S 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,
%T 35,36,37,38,40,41,42,43,45,46,47,48,50,51,52,53,54,56,57,58,59,61,62,
%U 63,64,65,67,68,69,70,72,73,74,75,76,78,79,80,81,83,84,85,86,87,89,90
%N Minimal positive number m for which Sum_{k=1..m} k^n < m!.
%C a(n) > n, with a(n)/n -> 1 as n -> infinity. - _Robert Israel_, May 28 2018
%H Robert Israel, <a href="/A113777/b113777.txt">Table of n, a(n) for n = 0..7485</a>
%F Let S(n, m) = Sum_{k=1..m} k^n. Define a(n) = min{ m | S(n, m)<m! }.
%e 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.
%p f:= proc(n) local k,L;
%p L:= sum(k^n,k=1..n);
%p for k from n+1 do
%p L:= L + k^n;
%p if L < k! then return k fi
%p od
%p end proc:
%p map(f, [$0..100]); # _Robert Israel_, May 28 2018
%o (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
%K nonn
%O 0,1
%A _Hieronymus Fischer_, Jan 19 2006
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