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%I #18 May 23 2023 07:29:27
%S 0,1,2,3,4,5,6,6,7,8,9,10,11,12,13,14,15,16,17,17,18,19,20,21,22,23,
%T 24,25,26,27,28,29,30,30,31,32,33,34,35,36,37,38,39,40,41,42,43,43,44,
%U 45,46,47,48,49,50,51,52,53,54,55,56,57,58,58,59,60,61,62,63,64,65,66,67
%N Smallest k such that n*(n+1)*(n+2)...*(n+k) >= n^n.
%C Limit_{n -> oo} a(n)/n = 1. [edited by _Robert Israel_, Nov 09 2022]
%H Robert Israel, <a href="/A076946/b076946.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) ~ n. - _Charles R Greathouse IV_, Nov 17 2022
%e a(3) = 2 as 3*4*5 > 3^3. but 3*4 < 3^3.
%p f:= proc(n) local k,s,t;
%p t:= n^n;
%p s:= 1;
%p for k from 0 do
%p s:= s*(n+k);
%p if s >= t then return k fi
%p od;
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, Nov 09 2022
%t a[n_] := Module[{k, nn = n^n, p = 1}, For[k = 0, True, k++, p = p(n+k); If[p >= nn, Return[k]]]];
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, May 23 2023 *)
%o (PARI) a(n) = my(x=n^n, p=1); for (k=0, oo, p*=(n+k); if (p>=x, return(k))); \\ _Michel Marcus_, Nov 17 2022
%K nonn
%O 1,3
%A _Amarnath Murthy_, Oct 20 2002
%E Edited, corrected and extended by _Robert G. Wilson v_, Oct 21 2002
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