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

1,4

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

Links

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

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.

Example

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

Mathematica

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

Prog

(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

Crossrefs

Sequence in context: A189663 A341440 A355028 * A029920 A100719 A057354

Adjacent sequences: A061372 A061373 A061374 * A061376 A061377 A061378

Keyword

nonn

Author

Amarnath Murthy, May 02 2001

Extensions

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

Status

approved