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

1,2

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).

Links

Amiram Eldar, Table of n, a(n) for n = 1..2970

Graeme McRae, Evil and Odious Numbers, Factorial, and Superfactorial, Maths Blog, Quora.

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A083784 A177048 A160819 * A205441 A181350 A174111

Adjacent sequences: A299697 A299698 A299699 * A299701 A299702 A299703

Keyword

nonn

Author

Graeme McRae, Feb 17 2018

Status

approved