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 1-5 of 5 results.

A056039 Largest k such that (k!)^2 divides n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 15, 15, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 28, 28, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37
Offset: 1

Views

Author

Labos Elemer, Jul 25 2000

Keywords

Comments

Let d(n) = a(n) - floor(n/2). Then, d(n) >= 0, and below 1000, d=1 arises 93 times, and d=2 arises 4 times.
The first occurrence of d(k) = 0, 1, 2, ..., is at k = 2, 1 (=A056067(1)), 416 (=A056068(1)), 6950, 16348, 505930, ... . - Amiram Eldar, May 24 2024

Links

Crossrefs

Cf. A000142, A001057, A001405, A055772, A056038, A056067, A056068.
Cf. A001044, A105350.

Programs

  • Mathematica
    a[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; k - 1]; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

Formula

A105350(n) = A001044(a(n)).

A056067 Numbers k such that k! is divisible by the square of (f+d)!^2 for d=0 and d=1 (and possibly larger d), where f = floor(k/2).

Original entry on oeis.org

1, 10, 11, 28, 29, 54, 55, 82, 83, 88, 89, 130, 131, 152, 153, 180, 181, 218, 219, 250, 251, 278, 279, 304, 305, 310, 311, 338, 339, 372, 373, 378, 379, 406, 407, 416, 417, 418, 419, 438, 439, 454, 455, 460, 461, 474, 475, 530, 531, 550, 551, 596, 597, 614
Offset: 1

Views

Author

Labos Elemer, Jul 26 2000

Keywords

Comments

Observe that all terms (except 1) are pairs of consecutive numbers starting with an even number (e.g., 88, 89).
Numbers k such that A056039(k) > floor(k/2). - Amiram Eldar, May 24 2024

Examples

			For n = 10 and 11, 10! and 11! are both divisible by 5!^2 and 6!^2.

Links

Crossrefs

Cf. A056038, A056039, A056042, A056043, A056044.
A056068 is a subsequence.

Programs

  • Mathematica
    q[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; k - 1] > Floor[n/2]; Select[Range[620], q] (* Amiram Eldar, May 24 2024 *)

A056068 Numbers k such that k! is divisible by the square of (f+d)!^2 for d = 0, 1 and 2 (and possibly larger d), where f = floor(k/2).

Original entry on oeis.org

416, 417, 916, 917, 1974, 1975, 2440, 2441, 2910, 2911, 3194, 3195, 3778, 3779, 4024, 4025, 4288, 4289, 4660, 4661, 4954, 4955, 5326, 5327, 5982, 5983, 6706, 6707, 6830, 6831, 6860, 6861, 6878, 6879, 6950, 6951, 6952, 6953, 7102, 7103, 7126, 7127
Offset: 1

Views

Author

Labos Elemer, Jul 26 2000

Keywords

Comments

Numbers k such that A056039(k) > floor(k/2) + 1. - Amiram Eldar, May 24 2024

Links

Crossrefs

Cf. A056038, A056039, A056042, A056043, A056044.
Subsequence of A056067.

Programs

  • Mathematica
    q[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; k - 1] > Floor[n/2] + 1; Select[Range[7200], q] (* Amiram Eldar, May 24 2024 *)

A056044 Let k be the largest number such that k^2 divides n! and let m be the largest number such that m!^2 divides n!; a(n) = k/m!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 2, 6, 3, 3, 2, 2, 2, 2, 2, 2, 2, 10, 10, 30, 2, 2, 12, 12, 3, 3, 6, 30, 10, 10, 10, 30, 6, 6, 2, 2, 2, 30, 60, 60, 30, 210, 42, 42, 42, 42, 1, 1, 2, 2, 4, 4, 4, 4, 4, 84, 21, 21, 14, 14, 14, 42, 6, 6, 2, 2, 2, 10, 10, 70, 140, 140, 14, 126, 3, 3, 6, 30
Offset: 1

Views

Author

Labos Elemer, Jul 25 2000

Keywords

Examples

			For n = 11, 11! = 6! * 6! * 77, so A000188(11!) = A056038(11) = 6! and a(11) = 6!/6! = 1.

Links

Crossrefs

Cf. A000188, A055772, A001405, A001057, A056038.

Programs

  • Mathematica
    f[p_, e_] := p^Floor[e/2]; b[1] = 1; b[n_] := Times @@ f @@@ FactorInteger[n!];
c[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; (k-1)!];
a[n_] := b[n] / c[n]; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

Formula

a(n) = A000188(n!)/A056038(n).

Extensions

Name corrected by Amiram Eldar, May 24 2024

A105350 Largest squared factorial dividing n!.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 36, 36, 576, 576, 518400, 518400, 518400, 518400, 25401600, 25401600, 1625702400, 1625702400, 131681894400, 131681894400, 13168189440000, 13168189440000, 1593350922240000, 1593350922240000, 229442532802560000, 229442532802560000
Offset: 0

Views

Author

Reinhard Zumkeller, Apr 01 2005

Keywords

Comments

a(n) = A001044(A056039(n)) = A056038(n)^2.
Whenever n > 1 is not in A056067, a(n) = A180064(n). - Andrey Zabolotskiy, Oct 19 2023

Links

Crossrefs

Cf. A000142, A000290, A055071, A055772, A056067.
Cf. A180064.

Programs

  • Mathematica
    a[n_] := (For[k = 1, Divisible[n!, k!^2], k++]; (k-1)!^2)
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 07 2018 *)

Extensions

Data and offset corrected by Jean-François Alcover, Aug 07 2018
Edited by Andrey Zabolotskiy, Oct 18 2023
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.