A248771 -id:A248771 - cp's OEIS frontend

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

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).

A248770 Greatest 6th power integer that divides n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 64, 64, 64, 64, 64, 64, 64, 46656, 2985984, 2985984, 2985984, 2985984, 191102976, 191102976, 191102976, 191102976, 191102976, 2985984000000, 2985984000000, 2176782336000000, 139314069504000000, 139314069504000000, 139314069504000000

Offset: 1

Clark Kimberling, Oct 14 2014

Every term divides all its successors.

			a(8) = 64 because 64 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. A248771, 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 *)

a(n) = n!/A248772(n).

A248783 Number of integers k^6 that divide n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 8, 8, 8, 8, 8, 16, 16, 24, 30, 30, 30, 30, 36, 36, 36, 36, 36, 36, 36, 48, 56, 56, 112, 112, 112, 112, 128, 128, 128, 128, 192, 192, 216, 216, 270, 270, 270, 270, 300, 300, 300, 300, 300, 360, 396, 396

Offset: 1

Clark Kimberling, Oct 15 2014

			a(16) counts these divisors of 16!:  1^6, 2^6, 2^12, 3^6, 6^6, 12^6.

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

Cf. A000142, A055993, A248780, A248781, A248782, A248771.

z = 130; m = 6;
f[n_] := f[n] = FactorInteger[n!]; r[x_] := r[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]]}];
a[n_] := Apply[Times, 1 + r[v[n]]/m]
t = Table[a[n], {n, 1, z}] (* A248783 *)

a(n)=c=0;d=divisors(n!);for(i=1,#d,if(ispower(d[i])&&ispower(d[i])%6==0,c++));c+1
n=1;while(n<50,print1(a(n),", ");n++) \\ Derek Orr, Oct 20 2014

cp's OEIS Frontend

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

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A248770 Greatest 6th power integer that divides n!.

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A248783 Number of integers k^6 that divide n!.

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