login
A145642
Cubefree part of n!.
7
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
1,2
LINKS
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.
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)
MATHEMATICA
CubefreePart[n_Integer?Positive] := Times @@ Power @@@ ({#[[1]], Mod[ #[[2]], 3]} & /@ FactorInteger[n]); Table[CubefreePart[n! ], {n, 1, 40}]
PROG
(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
(PARI) a(n) = my(f=factor(n!)); f[, 2] = apply(x->(x % 3), f[, 2]); factorback(f); \\ Michel Marcus, Jan 06 2019
CROSSREFS
Cf. A000142 (n!), A050985 (cubefree part), A248762, A248763.
CF. A055204 (squarefree part of n!).
Sequence in context: A302033 A337643 A071301 * A056195 A083169 A276817
KEYWORD
nonn
AUTHOR
Artur Jasinski, Oct 15 2008
STATUS
approved