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A248762 Greatest cube that divides n!.

%I A248762 #21 Sep 01 2024 09:37:26
%S A248762 1,1,1,8,8,8,8,64,1728,1728,1728,13824,13824,13824,46656000,
%T A248762 2985984000,2985984000,2985984000,2985984000,23887872000,
%U A248762 221225582592000,221225582592000,221225582592000,1769804660736000,221225582592000000,221225582592000000
%N A248762 Greatest cube that divides n!.
%C A248762 Every term divides all its successors.
%H A248762 Clark Kimberling, <a href="/A248762/b248762.txt">Table of n, a(n) for n = 1..1000</a>
%F A248762 a(n) = n!/A145642(n).
%F A248762 From _Amiram Eldar_, Sep 01 2024: (Start)
%F A248762 a(n) = A008834(n!).
%F A248762 a(n) = A248763(n)^3. (End)
%e A248762 a(4) = 8 because 8 divides 24 and if k > 2 then k^3 does not divide 24.
%t A248762 z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
%t A248762 u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
%t A248762 v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
%t A248762 p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
%t A248762 m = 3; Table[p[m, n], {n, 1, z}]  (* A248762 *)
%t A248762 Table[p[m, n]^(1/m), {n, 1, z}]   (* A248763 *)
%t A248762 Table[n!/p[m, n], {n, 1, z}]      (* A145642 *)
%t A248762 gk[n_]:=Select[Divisors[n!],IntegerQ[Surd[#,3]]&]; Max[#]&/@Array[gk,30] (* _Harvey P. Dale_, Sep 16 2021 *)
%t A248762 f[p_, e_] := p^(3*Floor[e/3]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30] (* _Amiram Eldar_, Sep 01 2024 *)
%o A248762 (PARI) a(n)=k=ceil((n!/2)^(1/3));while(n!%k^3,k--);k^3
%o A248762 vector(20,n,a(n)) \\ _Derek Orr_, Oct 19 2014
%o A248762 (PARI) a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(3*(f[i, 2]\3)));} \\ _Amiram Eldar_, Sep 01 2024
%Y A248762 Cf. A000142, A008834, A145642, A248763.
%K A248762 nonn,easy
%O A248762 1,4
%A A248762 _Clark Kimberling_, Oct 14 2014