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A374989 Square root of largest unitary square divisor of n!.

%I A374989 #14 Jul 27 2024 12:33:09
%S A374989 1,1,1,1,1,1,12,12,3,9,720,720,160,160,35,189,189,189,145152,145152,
%T A374989 7257600,12800,275,275,136857600,684288000,4343625,17875,875875,
%U A374989 875875,125536739328,125536739328,15324309,637,709689344,9052160000,18104320000,18104320000,2624375
%N A374989 Square root of largest unitary square divisor of n!.
%C A374989 Unitary analog of A055772.
%C A374989 a(n) is even if and only if n > 1 and is in A006364.
%H A374989 Amiram Eldar, <a href="/A374989/b374989.txt">Table of n, a(n) for n = 0..1118</a>
%F A374989 a(n) = sqrt(A374988(n)).
%F A374989 a(n) = A071974(n!).
%t A374989 f[p_, e_] := If[EvenQ[e], p^(e/2), 1]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40, 0]
%o A374989 (PARI) a(n) = {my(f = factor(n!)); prod(i = 1, #f~, if(f[i, 2]%2, 1, f[i, 1]^(f[i, 2]/2)));}
%o A374989 (Python)
%o A374989 from math import prod
%o A374989 from itertools import count, islice
%o A374989 from collections import Counter
%o A374989 from sympy import factorint
%o A374989 def A374989_gen(): # generator of terms
%o A374989     c = Counter()
%o A374989     for i in count(0):
%o A374989         c += Counter(factorint(i))
%o A374989         yield prod(p**(e>>1) for p, e in c.items() if e&1^1)
%o A374989 A374989_list = list(islice(A374989_gen(),30)) # _Chai Wah Wu_, Jul 27 2024
%Y A374989 Cf. A006364, A055772, A071974, A374988.
%K A374989 nonn
%O A374989 0,7
%A A374989 _Amiram Eldar_, Jul 26 2024