A145642 - OEIS

%I #26 Sep 17 2024 15:17:14

%S 1,2,6,3,15,90,630,630,210,2100,23100,34650,450450,6306300,28028,7007,

%T 119119,2144142,40738698,101846745,230945,5080790,116858170,350574510,

%U 70114902,1822987452,1822987452,6380456082,185033226378,5550996791340

%N Cubefree part of n!.

%H Chai Wah Wu, <a href="/A145642/b145642.txt">Table of n, a(n) for n = 1..500</a>

%H Rafael Jakimczuk, <a href="https://doi.org/10.12988/imf.2017.7542">On the h-th free part of the factorial</a>, International Mathematical Forum, Vol. 12, No. 13 (2017), pp. 629-634.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubefreePart.html">Cubefree Part</a>.

%F a(n) = A050985(A000142(n)). - _Michel Marcus_, Nov 07 2013

%F From _Amiram Eldar_, Sep 01 2024: (Start)

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

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

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

%o (Python)

%o from operator import mul

%o from functools import reduce

%o from sympy import factorint

%o import math

%o def A145642(n):

%o     return 1 if n <=1 else reduce(mul,[p**(e % 3) for p,e in factorint(math.factorial(n)).items()])

%o # _Chai Wah Wu_, Feb 04 2015

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

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

%Y CF. A055204 (squarefree part of n!).

%K nonn

%O 1,2

%A _Artur Jasinski_, Oct 15 2008