A056043 Let k be largest number such that k^2 divides n!; a(n) = k/floor(n/2)!.

1, 1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 2, 2, 2, 6, 3, 3, 2, 2, 2, 2, 2, 2, 2, 10, 10, 30, 30, 30, 12, 12, 3, 3, 6, 30, 10, 10, 10, 30, 6, 6, 2, 2, 2, 30, 60, 60, 30, 210, 42, 42, 42, 42, 28, 28, 2, 2, 4, 4, 4, 4, 4, 84, 21, 21, 14, 14, 14, 42, 6, 6, 2, 2, 2, 10, 10, 70, 140, 140, 14, 126, 126

Offset: 1

Author

Labos Elemer, Jul 25 2000

Keywords

• nonn

Examples

			For n = 7, 7! = 5040 = 144*35, so 12 is its largest square-root-divisor, A000188(5040), and it is divisible by 6 = 3!, so a(7) = 12/3! = 2.
		

Crossrefs

Cf. A000142, A000188, A001405, A004526, A055772, A081123.

Programs

Mathematica

f[p_, e_] := p^Floor[e/2]; b[1] = 1; a[n_] := (Times @@ f @@@ FactorInteger[n!]) / Floor[n/2]!; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

Formula

a(n) = A000188(n!)/floor(n/2)! = A055772(n)/A000142(A004526(n)) = A055772(n)/A081123(n). [Corrected by Amiram Eldar, May 24 2024]