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

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

Author

Labos Elemer, Jul 25 2000

Keywords

• nonn

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

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

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)).