A299700 Squarefree part of 1!*2!*3!*...*n!: The product of factorials one through n divided by its largest square divisor.

1, 2, 3, 2, 15, 3, 105, 6, 105, 15, 1155, 5, 15015, 70, 1001, 70, 17017, 35, 323323, 7, 138567, 154, 3187041, 231, 3187041, 6006, 1062347, 858, 30808063, 715, 955049953, 1430, 260468169, 12155, 9116385915, 12155, 337306278855, 461890, 8648878945, 46189, 354604036745, 1939938, 15247973580035, 176358

Offset: 1

Author

Graeme McRae, Feb 17 2018

Keywords

• nonn

Comments

Smallest number such that a(n)*1!*2!*3!*...*n! is a square.

If n is even, a(2n) = A055204(n).

If n is odd and evil (A129771) then a(2n) = A055204(n)/2.

If n is odd and odious (A092246) then a(2n) = 2*A055204(n).

Examples

			1!*2!*3!*4!*5! = 2^8 * 3^3 * 5^1 so a(5) = 3*5 = 15.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2970
• Graeme McRae, Evil and Odious Numbers, Factorial, and Superfactorial, Maths Blog, Quora.

Crossrefs

Cf. A000178, A007913, A055204, A092246, A129771.

Programs

Mathematica

Nest[Append[#, {#, Sqrt[#] /. (c_: 1) a_^(b_: 0) :> (c a^b)^2} &[#[[-1, 1]]*Length[# + 1]!]] &, {{1, 1}}, 44][[All, -1]] (* Michael De Vlieger, Feb 17 2018, after Bill Gosper at A007913 *)
f[n_] := Block[{m = BarnesG[n +2], p = 2}, While[p < n, While[ Mod[m, p^2] == 0, m/=p^2]; p = NextPrime@ p]; m]; Array[f, 42] (* Robert G. Wilson v, Feb 18 2018 *)

PARI

a(n) = core(prod(k=1, n, k!)); \\ Michel Marcus, Feb 17 2018

Formula

a(n) = A007913(A000178(n)). - Michel Marcus, Feb 17 2018