cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Clark Kimberling, Oct 14 2014

Keywords

Comments

Every term divides all its successors.

Examples

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

Links

Crossrefs

Cf. A248777, A248778, A000142.

Programs

  • Mathematica
    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 *)

Formula

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

A248777 Greatest k such that k^8 divides n!

Original entry on oeis.org

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, 24, 24, 24, 24, 24, 48, 240, 720, 720, 720, 720, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 10080, 10080, 10080, 20160, 20160, 60480, 60480, 60480, 60480
Offset: 1

Views

Author

Clark Kimberling, Oct 14 2014

Keywords

Comments

Every term divides all its successors.

Examples

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

Links

Crossrefs

Cf. A248776, A248778, A000142.

Programs

  • Mathematica
    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 *)

A248779 Rectangular array, by antidiagonals: T(m,n) = greatest (m+1)-th-power-free divisor of n!.

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 6, 6, 2, 1, 30, 3, 6, 2, 1, 5, 15, 24, 6, 2, 1, 35, 90, 120, 24, 6, 2, 1, 70, 630, 45, 120, 24, 6, 2, 1, 70, 630, 315, 720, 120, 24, 6, 2, 1, 7, 210, 2520, 5040, 720, 120, 24, 6, 2, 1, 77, 2100, 280, 1260, 5040, 720, 120, 24, 6, 2, 1, 231
Offset: 1

Views

Author

Clark Kimberling, Oct 14 2014

Keywords

Comments

Row 1: A055204, greatest squarefree divisor of n!
Row 2: A145642, greatest cubefree divisor of n!
Row 3: A248766, greatest 4th-power-free divisor of n!
Rows 4 to 7: A248769, A248772, A248775, A248778.
(The divisors are here called "greatest" rather than "largest" because the name refers to ">", called "greater than".)

Examples

			Northwest corner:
1   2   6   6   30   5    35    70
1   2   6   3   15   90   630   630
1   2   6   24  120  45   315   2520
1   2   6   24  120  720  5040  1260

Links

Crossrefs

Cf. A055204, A145642, A248766, A248769, A248772, A248775, A248778, A000142.

Programs

  • Mathematica
    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]]}]
t = Table[n!/p[m, n], {m, 2, 16}, {n, 1, 16}]; TableForm[t]  (* A248779 array *)
f = Table[t[[n - k + 1, k]], {n, 12}, {k, n, 1, -1}] // Flatten (* A248779 seq. *)
Showing 1-3 of 3 results.

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