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A145642 Cubefree part of n!.

%I A145642 #27 Feb 16 2025 08:33:09
%S A145642 1,2,6,3,15,90,630,630,210,2100,23100,34650,450450,6306300,28028,7007,
%T A145642 119119,2144142,40738698,101846745,230945,5080790,116858170,350574510,
%U A145642 70114902,1822987452,1822987452,6380456082,185033226378,5550996791340
%N A145642 Cubefree part of n!.
%H A145642 Chai Wah Wu, <a href="/A145642/b145642.txt">Table of n, a(n) for n = 1..500</a>
%H A145642 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 A145642 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CubefreePart.html">Cubefree Part</a>.
%F A145642 a(n) = A050985(A000142(n)). - _Michel Marcus_, Nov 07 2013
%F A145642 From _Amiram Eldar_, Sep 01 2024: (Start)
%F A145642 a(n) = n! / A248762(n) = n! / A248763(n)^3.
%F A145642 log(a(n)) = log(3) * n + o(n) (Jakimczuk, 2017). (End)
%t A145642 CubefreePart[n_Integer?Positive] := Times @@ Power @@@ ({#[[1]], Mod[ #[[2]], 3]} & /@ FactorInteger[n]); Table[CubefreePart[n! ], {n, 1, 40}]
%o A145642 (Python)
%o A145642 from operator import mul
%o A145642 from functools import reduce
%o A145642 from sympy import factorint
%o A145642 import math
%o A145642 def A145642(n):
%o A145642     return 1 if n <=1 else reduce(mul,[p**(e % 3) for p,e in factorint(math.factorial(n)).items()])
%o A145642 # _Chai Wah Wu_, Feb 04 2015
%o A145642 (PARI) a(n) = my(f=factor(n!)); f[,2] = apply(x->(x % 3), f[,2]); factorback(f); \\ _Michel Marcus_, Jan 06 2019
%Y A145642 Cf. A000142 (n!), A050985 (cubefree part), A248762, A248763.
%Y A145642 CF. A055204 (squarefree part of n!).
%K A145642 nonn
%O A145642 1,2
%A A145642 _Artur Jasinski_, Oct 15 2008