%I #6 Mar 12 2015 20:33:04

%S 1,2,6,24,120,720,5040,40320,362880,14175,155925,1871100,24324300,

%T 340540200,5108103000,81729648000,1389404016000,14889875,282907625,

%U 5658152500,118821202500,2614066455000,60123528465000,1442964683160000,36074117079000000

%N Greatest 8th-power-free divisor of n!.

%H Clark Kimberling, <a href="/A248778/b248778.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%t z = 60; 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 = 8; Table[p[m, n], {n, 1, z}] (* A248776 *)

%t Table[p[m, n]^(1/m), {n, 1, z}] (* A248777 *)

%t Table[n!/p[m, n], {n, 1, z}] (* A248778 *)

%Y Cf. A248776, A248777, A000142.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Oct 14 2014