A248768 -id:A248768 - cp's OEIS frontend

A248769 Greatest 5th-power-free divisor of n!.

1, 2, 6, 24, 120, 720, 5040, 1260, 11340, 113400, 1247400, 1925, 25025, 350350, 5255250, 2627625, 44669625, 804053250, 15277011750, 305540235000, 6416344935000, 141159588570000, 3246670537110000, 10020588077500, 80164704620, 2084282320120, 56275622643240

Offset: 1

Clark Kimberling, Oct 14 2014

			a(8) = 1260 because 1260 divides 8! and if k > 1260 divides 8!, then h^5 divides 8!/k for some h > 1.

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

Cf. A248767, A248768, 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 = 5; Table[p[m, n], {n, 1, z}]  (* A248767 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248768 *)
Table[n!/p[m, n], {n, 1, z}]      (* A248769 *)

a(n) = n!/A248767(n).

A248767 Greatest 5th power integer that divides n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 32, 32, 32, 32, 248832, 248832, 248832, 248832, 7962624, 7962624, 7962624, 7962624, 7962624, 7962624, 7962624, 7962624, 61917364224, 193491763200000, 193491763200000, 193491763200000, 6191736422400000, 6191736422400000, 6191736422400000

Offset: 1

Clark Kimberling, Oct 14 2014

Every term divides all its successors.

			a(8) = 32 because 32 divides 8! and if k > 2 then k^5 does not divide 8!.

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

Cf. A248768, A248769, 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 = 5; Table[p[m, n], {n, 1, z}]  (* A248767 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248768 *)
Table[n!/p[m, n], {n, 1, z}]      (* A248769 *)

a(n) = n!/A248769(n).

A248782 Number of integers k^5 that divide n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 15, 30, 30, 30, 36, 36, 36, 36, 42, 56, 56, 112, 112, 112, 128, 128, 128, 128, 128, 128, 144, 270, 270, 270, 300, 300, 300, 300, 300, 300, 396, 792, 792, 792, 792, 792, 864, 864, 864, 1512

Offset: 1

Clark Kimberling, Oct 15 2014

			a(12) counts these divisors of 12!:  1, 32, 243, 1024, 7776, 248832.

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

Cf. A000142, A055993, A248780, A248781, A248768.

z = 130; m = 5;
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}] (* A248782 *)

a(n)=c=0;d=divisors(n!);for(i=1,#d,if(ispower(d[i])&&ispower(d[i])%5==0,c++));c+1
n=1;while(n<50,print1(a(n),", ");n++) \\ Derek Orr, Oct 20 2014

cp's OEIS Frontend

A248769 Greatest 5th-power-free divisor of n!.

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A248767 Greatest 5th power integer that divides n!.

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A248782 Number of integers k^5 that divide n!.

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