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%I #16 Sep 16 2021 12:35:35
%S 1,1,1,8,8,8,8,64,1728,1728,1728,13824,13824,13824,46656000,
%T 2985984000,2985984000,2985984000,2985984000,23887872000,
%U 221225582592000,221225582592000,221225582592000,1769804660736000,221225582592000000,221225582592000000
%N Greatest cube that divides n!.
%C Every term divides all its successors.
%H Clark Kimberling, <a href="/A248762/b248762.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = n!/A145642(n).
%e a(4) = 8 because 8 divides 24 and if k > 2 then k^3 does not divide 24.
%t z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
%t u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
%t v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
%t p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
%t m = 3; Table[p[m, n], {n, 1, z}] (* A248762 *)
%t Table[p[m, n]^(1/m), {n, 1, z}] (* A248763 *)
%t Table[n!/p[m, n], {n, 1, z}] (* A145642 *)
%t gk[n_]:=Select[Divisors[n!],IntegerQ[Surd[#,3]]&]; Max[#]&/@Array[gk,30] (* _Harvey P. Dale_, Sep 16 2021 *)
%o (PARI) a(n)=k=ceil((n!/2)^(1/3));while(n!%k^3,k--);k^3
%o vector(20,n,a(n)) \\ _Derek Orr_, Oct 19 2014
%Y Cf. A248763, A145642, A000142.
%K nonn,easy
%O 1,4
%A _Clark Kimberling_, Oct 14 2014
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