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

1,2

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.

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

Adjacent sequences: A145639 A145640 A145641 * A145643 A145644 A145645

Keyword

nonn

Author

Artur Jasinski, Oct 15 2008

Status

approved