A248778 Greatest 8th-power-free divisor of n!.

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 14175, 155925, 1871100, 24324300, 340540200, 5108103000, 81729648000, 1389404016000, 14889875, 282907625, 5658152500, 118821202500, 2614066455000, 60123528465000, 1442964683160000, 36074117079000000

Offset: 1

Author

Clark Kimberling, Oct 14 2014

Keywords

• nonn
• easy

Examples

			a(8) = 315 because 315 divides 8! and if k > 315 divides 8!, then h^7 divides 8!/k for some h > 1.
		

Links

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

Crossrefs

Cf. A248776, A248777, A000142.

Programs

Mathematica

z = 60; 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 = 8; Table[p[m, n], {n, 1, z}]  (* A248776 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248777 *)
Table[n!/p[m, n], {n, 1, z}]      (* A248778 *)

Formula

a(n) = n!/A248773(n).