Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #37 Sep 06 2020 03:47:07
%S 1,2,3,2,15,3,105,6,105,15,1155,5,15015,70,1001,70,17017,35,323323,7,
%T 138567,154,3187041,231,3187041,6006,1062347,858,30808063,715,
%U 955049953,1430,260468169,12155,9116385915,12155,337306278855,461890,8648878945,46189,354604036745,1939938,15247973580035,176358
%N Squarefree part of 1!*2!*3!*...*n!: The product of factorials one through n divided by its largest square divisor.
%C Smallest number such that a(n)*1!*2!*3!*...*n! is a square.
%C If n is even, a(2n) = A055204(n).
%C If n is odd and evil (A129771) then a(2n) = A055204(n)/2.
%C If n is odd and odious (A092246) then a(2n) = 2*A055204(n).
%H Amiram Eldar, <a href="/A299700/b299700.txt">Table of n, a(n) for n = 1..2970</a>
%H Graeme McRae, <a href="https://graemesmathblog.quora.com/Evil-and-Odious-Numbers-Factorial-and-Superfactorial">Evil and Odious Numbers, Factorial, and Superfactorial</a>, Maths Blog, Quora.
%F a(n) = A007913(A000178(n)). - _Michel Marcus_, Feb 17 2018
%e 1!*2!*3!*4!*5! = 2^8 * 3^3 * 5^1 so a(5) = 3*5 = 15.
%t 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 *)
%t 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 *)
%o (PARI) a(n) = core(prod(k=1, n, k!)); \\ _Michel Marcus_, Feb 17 2018
%Y Cf. A000178, A007913, A055204, A092246, A129771.
%K nonn
%O 1,2
%A _Graeme McRae_, Feb 17 2018