A145642 Cubefree part of n!.

1, 2, 6, 3, 15, 90, 630, 630, 210, 2100, 23100, 34650, 450450, 6306300, 28028, 7007, 119119, 2144142, 40738698, 101846745, 230945, 5080790, 116858170, 350574510, 70114902, 1822987452, 1822987452, 6380456082, 185033226378, 5550996791340

Offset: 1

Author

Artur Jasinski, Oct 15 2008

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..500
• Rafael Jakimczuk, On the h-th free part of the factorial, International Mathematical Forum, Vol. 12, No. 13 (2017), pp. 629-634.
• Eric Weisstein's World of Mathematics, Cubefree Part.

Crossrefs

Cf. A000142 (n!), A050985 (cubefree part), A248762, A248763.

CF. A055204 (squarefree part of n!).

Programs

Mathematica

CubefreePart[n_Integer?Positive] := Times @@ Power @@@ ({#[[1]], Mod[ #[[2]], 3]} & /@ FactorInteger[n]); Table[CubefreePart[n! ], {n, 1, 40}]

PARI

a(n) = my(f=factor(n!)); f[,2] = apply(x->(x % 3), f[,2]); factorback(f); \\ Michel Marcus, Jan 06 2019

Python

from operator import mul
from functools import reduce
from sympy import factorint
import math
def A145642(n):
    return 1 if n <=1 else reduce(mul,[p**(e % 3) for p,e in factorint(math.factorial(n)).items()])
# Chai Wah Wu, Feb 04 2015

Formula

a(n) = A050985(A000142(n)). - Michel Marcus, Nov 07 2013

From Amiram Eldar, Sep 01 2024: (Start)

a(n) = n! / A248762(n) = n! / A248763(n)^3.

log(a(n)) = log(3) * n + o(n) (Jakimczuk, 2017). (End)