A248770 -id:A248770 - cp's OEIS frontend

A248771 Greatest k such that k^6 divides n!

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 6, 12, 12, 12, 12, 24, 24, 24, 24, 24, 120, 120, 360, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 4320, 8640, 8640, 60480, 60480, 60480, 60480, 120960, 120960, 120960, 120960, 604800

Offset: 1

Clark Kimberling, Oct 14 2014

Every term divides all its successors.

			a(8) = 2 because 2^6 divides 8! and if k > 2 then k^6 does not divide 8!.

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

Cf. A248770, A248772, A000142.

z = 50; 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 = 6; Table[p[m, n], {n, 1, z}]  (* A248770 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248771 *)
Table[n!/p[m, n], {n, 1, z}]      (* A248772 *)

A248772 Greatest 6th-power-free divisor of n!.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 630, 5670, 56700, 623700, 7484400, 97297200, 1362160800, 28028000, 7007000, 119119000, 2144142000, 40738698000, 12730843125, 267347705625, 5881649523750, 135277939046250, 3246670537110000, 5194672859376, 135061494343776

Offset: 1

Clark Kimberling, Oct 14 2014

			a(8) = 630 because 630 divides 8! and if k > 630 divides 8!, then h^6 divides 8!/k for some h > 1.

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

Cf. A248770, A248771, A000142.

z = 50; 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 = 6; Table[p[m, n], {n, 1, z}]  (* A248770 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248771 *)
Table[n!/p[m, n], {n, 1, z}]      (* A248772 *)

a(n) = n!/A248770(n).

cp's OEIS Frontend

A248771 Greatest k such that k^6 divides n!

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