A248762 Greatest cube that divides n!.

1, 1, 1, 8, 8, 8, 8, 64, 1728, 1728, 1728, 13824, 13824, 13824, 46656000, 2985984000, 2985984000, 2985984000, 2985984000, 23887872000, 221225582592000, 221225582592000, 221225582592000, 1769804660736000, 221225582592000000, 221225582592000000

Offset: 1

Author

Clark Kimberling, Oct 14 2014

Keywords

• nonn
• easy

Comments

Every term divides all its successors.

Examples

			a(4) = 8 because 8 divides 24 and if k > 2 then k^3 does not divide 24.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000142, A008834, A145642, A248763.

Programs

Mathematica

z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
m = 3; Table[p[m, n], {n, 1, z}]  (* A248762 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248763 *)
Table[n!/p[m, n], {n, 1, z}]      (* A145642 *)
gk[n_]:=Select[Divisors[n!],IntegerQ[Surd[#,3]]&]; Max[#]&/@Array[gk,30] (* Harvey P. Dale, Sep 16 2021 *)
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 *)

PARI

a(n)=k=ceil((n!/2)^(1/3));while(n!%k^3,k--);k^3
vector(20,n,a(n)) \\ Derek Orr, Oct 19 2014

PARI

a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(3*(f[i, 2]\3)));} \\ Amiram Eldar, Sep 01 2024

Formula

a(n) = n!/A145642(n).

From Amiram Eldar, Sep 01 2024: (Start)

a(n) = A008834(n!).

a(n) = A248763(n)^3. (End)