|
%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
|
