A105350 -id:A105350 - 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).

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

Labos Elemer, Jul 25 2000

nonn

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

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

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

Cf. A001044, A105350.

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

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

cp's OEIS Frontend

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

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