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.

A248765 Greatest k such that k^4 divides n!

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 6, 12, 12, 12, 12, 12, 12, 24, 24, 144, 144, 720, 720, 720, 720, 1440, 1440, 1440, 4320, 60480, 60480, 60480, 60480, 120960, 120960, 241920, 1209600, 3628800, 3628800, 3628800, 3628800, 7257600
Offset: 1

Views

Author

Clark Kimberling, Oct 14 2014

Keywords

Comments

Every term divides all its successors.

Examples

			a(6) = 2 because 2^4 divides 6! and if k > 2 then k^4 does not divide 6!.

Links

Crossrefs

Cf. A000142, A053164, A248762, A248764, A248766.

Programs

  • Mathematica
    z = 40; 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 = 4; Table[p[m, n], {n, 1, z}]  (* A248764 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248765 *)
Table[n!/p[m, n], {n, 1, z}]      (* A248766 *)
f[p_, e_] := p^Floor[e/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30] (* Amiram Eldar, Sep 01 2024 *)
  • PARI
    a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(f[i, 2]\4));} \\ Amiram Eldar, Sep 01 2024

Formula

From Amiram Eldar, Sep 01 2024: (Start)
a(n) = A053164(n!).
a(n) = (n! / A248766(n))^(1/4) = A248764(n)^(1/4).
log(a(n)) = (1/4)*n*log(n) - (2*log(2)+1)*n/4 + o(n) (Jakimczuk, 2017). (End)

A248764 Greatest 4th power integer that divides n!

Original entry on oeis.org

1, 1, 1, 1, 1, 16, 16, 16, 1296, 20736, 20736, 20736, 20736, 20736, 20736, 331776, 331776, 429981696, 429981696, 268738560000, 268738560000, 268738560000, 268738560000, 4299816960000, 4299816960000, 4299816960000, 348285173760000, 13379723235164160000
Offset: 1

Views

Author

Clark Kimberling, Oct 14 2014

Keywords

Comments

Every term divides all its successors.

Examples

			a(6) = 16 because 16 divides 6! and if k > 2 then k^4 does not divide 6!.

Links

Crossrefs

Cf. A000142, A008835, A248762, A248765, A248766.

Programs

  • Mathematica
    z = 40; 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 = 4; Table[p[m, n], {n, 1, z}]  (* A248764 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248765 *)
Table[n!/p[m, n], {n, 1, z}]      (* A248766 *)
f[p_, e_] := p^(4*Floor[e/4]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30] (* Amiram Eldar, Sep 01 2024 *)
  • PARI
    a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(4*(f[i, 2]\4)));} \\ Amiram Eldar, Sep 01 2024

Formula

a(n) = n!/A248766(n).
From Amiram Eldar, Sep 01 2024: (Start)
a(n) = A008835(n!).
a(n) = A248765(n)^4. (End)

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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