A248777 -id:A248777 - cp's OEIS frontend

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

Clark Kimberling, Oct 14 2014

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

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

Cf. A248776, A248777, A000142.

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 *)

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

A248776 Greatest 8th power integer that divides n!

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 256, 256, 256, 256, 256, 256, 256, 256, 429981696, 429981696, 429981696, 429981696, 429981696, 429981696, 429981696, 429981696, 429981696, 429981696, 110075314176, 110075314176, 110075314176, 110075314176, 110075314176

Offset: 1

Clark Kimberling, Oct 14 2014

Every term divides all its successors.

			a(8) = 128 because 128 divides 8! and if k > 2 then k^8 does not divide 8!.

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

Cf. A248777, A248778, A000142.

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 *)
Module[{e=Range[30]^8},Table[Max[Select[e,Divisible[n!,#]&]],{n,40}]] (* Harvey P. Dale, Dec 23 2019 *)

a(n) = n!/A248778(n).

cp's OEIS Frontend

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

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