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A306904
The geometric mean of the first n integers, rounded to the nearest integer.
1
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
1,3
COMMENTS
a(n) is the least k such that (k-1/2)^n < n!. - Robert Israel, May 05 2019
LINKS
FORMULA
a(n) = round(n!^(1/n)).
a(n) ~ n/e + log(n)/(2*e). - Robert Israel, May 05 2019
EXAMPLE
a(5) is the 5th root of the product of the first 5 integers (approx. 2.605171) rounded up to 3.
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 *)
PROG
(PARI) a(n) = round(n!^(1/n)); \\ Michel Marcus, May 05 2019
CROSSREFS
Cf. A000142, A061375 (floor instead of round), A214046.
Sequence in context: A077113 A143796 A245473 * A371627 A057362 A214046
KEYWORD
nonn
AUTHOR
Robin Powell, Mar 15 2019
EXTENSIONS
Corrected by Robert Israel, May 05 2019
STATUS
approved