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A299700 Squarefree part of 1!*2!*3!*...*n!: The product of factorials one through n divided by its largest square divisor. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 18 10:13 EDT 2021. Contains 347518 sequences. (Running on oeis4.)