A061375 - OEIS

%I #18 Jul 02 2018 01:41:17

%S 1,1,1,2,2,2,3,3,4,4,4,5,5,6,6,6,7,7,7,8,8,9,9,9,10,10,10,11,11,12,12,

%T 12,13,13,13,14,14,15,15,15,16,16,16,17,17,17,18,18,19,19,19,20,20,20,

%U 21,21,22,22,22,23,23,23,24,24,25,25,25,26,26,26,27

%N Integer part of geometric mean of first n positive integers.

%C a(n) < (n+1)/2 for n > 1 by the AM-GM (arithmetic-geometric mean) inequality.

%C Also, largest integer m such that m^n <= n!. - _M. F. Hasler_, Mar 07 2009

%H Harry J. Smith, <a href="/A061375/b061375.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = floor((n!)^(1/n)); (n!)^(1/n) ~ e^(-1) (n+log(2 Pi n)/2) + O(log(n)^2/n) as n -> infinity.

%e a(7) = floor((7!)^(1/7)) = floor(5040^(1/7)) = floor(3.380015...) = 3.

%t Table[IntegerPart[GeometricMean[Range[n]]],{n,80}] (* _Harvey P. Dale_, Mar 05 2014 *)

%o (PARI) A061375(n)=sqrtn(n!,n)\1 \\ _M. F. Hasler_, Mar 07 2009

%o (PARI) { default(realprecision, 100); f=1; for (n=1, 1000, f*=n; write("b061375.txt", n, " ", sqrtn(f, n)\1) ) } \\ _Harry J. Smith_, Jul 22 2009

%K nonn

%O 1,4

%A _Amarnath Murthy_, May 02 2001

%E Edited by _Franklin T. Adams-Watters_, May 18 2010