A061375 Integer part of geometric mean of first n positive integers.

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, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27

Offset: 1

Author

Amarnath Murthy, May 02 2001

Keywords

• nonn

Comments

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

Also, largest integer m such that m^n <= n!. - M. F. Hasler, Mar 07 2009

Examples

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

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Programs

Mathematica

Table[IntegerPart[GeometricMean[Range[n]]],{n,80}] (* Harvey P. Dale, Mar 05 2014 *)

PARI

A061375(n)=sqrtn(n!,n)\1 \\ M. F. Hasler, Mar 07 2009

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

Formula

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.

Extensions

Edited by Franklin T. Adams-Watters, May 18 2010