A056039 - OEIS

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

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OFFSET

1,4

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

FORMULA

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

MATHEMATICA

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