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

0,5

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.

Links

Table of n, a(n) for n=0..32.

Formula

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A132369 A282169 A081123 * A076929 A265642 A186944

Adjacent sequences: A056035 A056036 A056037 * A056039 A056040 A056041

Keyword

nonn

Author

Labos Elemer, Jul 25 2000

Status

approved