A306904 The geometric mean of the first n integers, rounded to the nearest integer.

1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25

Offset: 1

Author

Robin Powell, Mar 15 2019

Keywords

• nonn

Comments

a(n) is the least k such that (k-1/2)^n < n!. - Robert Israel, May 05 2019

Examples

			a(5) is the 5th root of the product of the first 5 integers (approx. 2.605171) rounded up to 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000142, A061375 (floor instead of round), A214046.

Programs

Maple

Res:= 1: last:= 1: v:= 1:
for n from 2 to 100 do
  v:= n*v; t:= 2^n*v;
  for k from last while (2*k-1)^n < t do od:
  last:= k-1;
  Res:= Res, last;
od:
Res; # Robert Israel, May 05 2019

Mathematica

Array[Round[#!^(1/#)] &, 68] (* Michael De Vlieger, Mar 31 2019 *)
Table[Round[GeometricMean[Range[n]]],{n,70}] (* Harvey P. Dale, Mar 19 2023 *)

PARI

a(n) = round(n!^(1/n)); \\ Michel Marcus, May 05 2019

Python

from sympy import integer_nthroot, factorial
def A306904(n): return (m:=int(integer_nthroot(f:=int(factorial(n)),n)[0]))+int((f<=((m<<1)+1)**n) # Chai Wah Wu, Jun 06 2025

Formula

a(n) = round(n!^(1/n)).

a(n) ~ n/e + log(n)/(2*e). - Robert Israel, May 05 2019

Extensions

Corrected by Robert Israel, May 05 2019