A056039 -id:A056039 - cp's OEIS frontend

A056038 Largest factorial k! such that (k!)^2 divides n!.

1, 1, 1, 1, 2, 2, 6, 6, 24, 24, 720, 720, 720, 720, 5040, 5040, 40320, 40320, 362880, 362880, 3628800, 3628800, 39916800, 39916800, 479001600, 479001600, 6227020800, 6227020800, 1307674368000, 1307674368000, 1307674368000, 1307674368000, 20922789888000

Offset: 0

Labos Elemer, Jul 25 2000

nonn

This is neither floor(n/2)! nor ceiling(n/2)!, but often coincides with one of them.

a(n) = k!, where k = floor(n/2) + d(n) and d = 0, 1, 2, ... . Below 1000, d = 1 arises 93 times, and d = 2 arises 4 times. See A056067 and A056068.

			For n = 10 or n = 11, floor(n/2)! = 5! = 120; 5!^2 = 14400 divides 10! = 14400*252 or 11! = 14400*2772. However, 10!/6!^2 = 7 and 11!/6!^2 = 77, i.e., (d + floor(n/2))^2 may divide n!. Here d = 1, but d > 1 also occurs as follows: for n = 416 or n = 417, floor(n/2) = 208, and 208!^2 divides 416! and 417!, but 209!^2 and 210!^2 also divide these factorials.

Cf. A000142, A001057, A001405, A055772, A056039, A056067, A056068, A105350.

a[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; (k-1)!]; Array[a, 33, 0] (* Amiram Eldar, May 24 2024 *)

a(n)^2 = A105350(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

Labos Elemer, Jul 26 2000

nonn

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A056038, A056039, A056042, A056043, A056044.

A056068 is a subsequence.

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

Labos Elemer, Jul 26 2000

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A056038, A056039, A056042, A056043, A056044.

Subsequence of A056067.

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 *)

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

Reinhard Zumkeller, Apr 01 2005

nonn

a(n) = A001044(A056039(n)) = A056038(n)^2.

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

Index entries for sequences related to factorial numbers.

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

Cf. A180064.

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 *)

Data and offset corrected by Jean-François Alcover, Aug 07 2018

Edited by Andrey Zabolotskiy, Oct 18 2023

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A056038 Largest factorial k! such that (k!)^2 divides n!.

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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).

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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).

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A105350 Largest squared factorial dividing n!.

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