|
%I #69 Dec 21 2019 16:27:27
%S 1,2,4,7,9,12,14,17,20,22,25,28,30,33,36,38,41,44,47,49,52,55,57,60,
%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
%N a(n) is the smallest k > 0 such that n^k < k!.
%C Differences are 1, 2 or 3 (see A065067). The limit as n -> infinity of a(n)/n is e. - _Robert G. Wilson v_, Dec 05 2001. [Apparently the Schonbek link contains a proof of the first assertion.]
%C a(10) = 25, a(100) = 269, a(1000) = 2714, a(10000) = 27177, a(10^5) = 271822, see A085830.
%C a(n) = least k such that geometric mean of {1,1/2,...,1/k} <= 1/n. - _Clark Kimberling_, Jul 11 2013
%C Let b(n) be the largest k such that n^k > k!. Then b(n) = a(n)-1. - _Joseph Damico_, Jun 30 2019
%H Harry J. Smith and Ely Golden, <a href="/A065027/b065027.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from Harry J. Smith)
%H Robert Israel, <a href="/A065027/a065027.png">Plot of a(n) - n e + log(sqrt(2 Pi n)) for 1 <= n <= 20000</a>
%H Tomas Schonbek, <a href="http://forumgeom.fau.edu/POLYA/ProblemCenter/POLYA008.html">POLYA008: First n for which m^n < n!</a> [From Nikos Apostolakis, Feb 17 2009]
%F It appears that L(n) < a(n) - n e + log(sqrt(2 Pi n)) < 1/2, where L(n) = -1/2 + o(1), and L(n) > -0.53 for all n. - _Robert Israel_, Oct 28 2016 (In other words, a(n) - n e + log(sqrt(2 Pi n)) < 1/2 for all n, and there is some function L(n) = -1/2 + o(1) such that -0.53 < L(n) < a(n) - n e + log(sqrt(2 Pi n)) for all n. - _Charles R Greathouse IV_, Nov 04 2016)
%e 2^3 > 3! but 2^4 < 4!, so a(2)=4.
%p m:= 1:
%p for n from 1 to 100 do
%p while n^m >= m! do m:=m+1 od:
%p A[n]:= m;
%p od:
%p seq(A[n],n=0..100); # _Robert Israel_, Oct 28 2016
%t Table[Length[Select[Table[m^n/n!,{n,1,180}],#>=1&]]+1,{m,1,61}]
%t sm0[n_]:=Module[{m=1},While[n^m>=m!,m++];m]; Array[sm0,70] (* _Harvey P. Dale_, Jan 24 2018 *)
%o (PARI) { m=1; for (n=1, 1000, until (n^m < m!, m++); write("b065027.txt", n, " ", m) ) } \\ _Harry J. Smith_, Oct 03 2009
%K easy,nonn
%O 0,2
%A _Floor van Lamoen_, Nov 02 2001
%E More terms from _Robert G. Wilson v_, Dec 05 2001
%E a(0)=1 prepended by _Alois P. Heinz_, Dec 21 2019
|
