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A353337 Number of ways to write n as a product of the terms of A028260 larger than 1; a(1) = 1 by convention (an empty product).

%I A353337 #16 Apr 19 2022 07:45:02
%S A353337 1,0,0,1,0,1,0,0,1,1,0,0,0,1,1,2,0,0,0,0,1,1,0,2,1,1,0,0,0,0,0,0,1,1,
%T A353337 1,3,0,1,1,2,0,0,0,0,0,1,0,0,1,0,1,0,0,2,1,2,1,1,0,3,0,1,0,3,1,0,0,0,
%U A353337 1,0,0,0,0,1,0,0,1,0,0,0,2,1,0,3,1,1,1,2,0,3,1,0,1,1,1,4,0,0,0,3,0,0,0,2,0
%N A353337 Number of ways to write n as a product of the terms of A028260 larger than 1; a(1) = 1 by convention (an empty product).
%C A353337 Number of factorizations of n into factors k > 1 for which there is an even number of primes (when counted with multiplicity, A001222) in their prime factorization.
%H A353337 Antti Karttunen, <a href="/A353337/b353337.txt">Table of n, a(n) for n = 1..65537</a>
%H A353337 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A353337 a(n) = a(A046523(n)). [The sequence depends only on the prime signature of n].
%F A353337 For all n >= 1, a(n) >= A320655(n), and a(n) >= A353377(n).
%e A353337 Of the eleven divisors of 96 larger than one, the following: [4, 6, 16, 24, 96] are terms of A028260 because they have an even number of prime factors when counted with repetition. Using them, we can factor 96 in four possible ways, as 96 = 24*4 = 16*6 = 6*4*4, therefore a(96) = 4.
%o A353337 (PARI)
%o A353337 A065043(n) = (1 - (bigomega(n)%2));
%o A353337 A353337(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A065043(d), s += A353337(n/d, d))); (s));
%Y A353337 Cf. A001222, A028260, A038548, A046523, A065043, A353338 [= a(n^2)].
%Y A353337 Cf. also A320655, A353377.
%K A353337 nonn
%O A353337 1,16
%A A353337 _Antti Karttunen_, Apr 17 2022