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A353377 Number of ways to write n as a product of the terms of A345452 larger than 1; a(1) = 1 by convention (an empty product).

%I A353377 #11 Apr 19 2022 11:41:40
%S A353377 1,0,0,1,0,0,0,0,1,0,0,0,0,0,1,2,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,
%T A353377 1,2,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,2,0,0,0,3,1,0,0,0,
%U A353377 1,0,0,0,0,0,0,0,1,0,0,0,2,0,0,2,1,0,1,0,0,0,1,0,1,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1
%N A353377 Number of ways to write n as a product of the terms of A345452 larger than 1; a(1) = 1 by convention (an empty product).
%C A353377 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, and the 2-adic valuation of k (A007814) is also even.
%H A353377 Antti Karttunen, <a href="/A353377/b353377.txt">Table of n, a(n) for n = 1..65537</a>
%F A353377 For all n >= 1, a(n) <= A353337(n).
%e A353377 Of the 19 divisors of 240 larger than 1, the following: [4, 15, 16, 60, 240] are found in A345452. Using them, we can factor 240 in four possible ways, as 240 = 60*4 = 16*15 = 15*4*4, therefore a(240) = 4.
%e A353377 Of the 23 divisors of 540 larger than 1, the following: [4, 9, 15, 36, 60, 135, 540] are found in A345452. Using them, we can factor 540 in five possible ways, as 540 = 135*4 = 60*9 = 36*15 = 15*9*4, therefore a(540) = 5.
%o A353377 (PARI)
%o A353377 A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2));
%o A353377 A353377(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353374(d), s += A353377(n/d, d))); (s));
%Y A353377 Cf. A001222, A007814, A345452, A353374, A353378 [= a(n^2)].
%Y A353377 Cf. also A353333, A353337, A353353, A353373.
%K A353377 nonn
%O A353377 1,16
%A A353377 _Antti Karttunen_, Apr 17 2022