login
Greatest k such that k^8 divides n!
3

%I #5 Oct 19 2014 16:22:46

%S 1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,12,12,12,12,12,12,12,12,12,12,24,

%T 24,24,24,24,24,48,240,720,720,720,720,720,720,720,720,1440,1440,1440,

%U 1440,1440,10080,10080,10080,20160,20160,60480,60480,60480,60480

%N Greatest k such that k^8 divides n!

%C Every term divides all its successors.

%H Clark Kimberling, <a href="/A248777/b248777.txt">Table of n, a(n) for n = 1..1000</a>

%e a(10) = 2 because 2^10 divides 8! and if k > 2 then k^8 does not divide 8!.

%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, A248778, A000142.

%K nonn,easy

%O 1,10

%A _Clark Kimberling_, Oct 14 2014