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

Author

Labos Elemer, Jul 25 2000

Keywords

• nonn

Comments

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.

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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

Formula

a(n)^2 = A105350(n).