A353377 - OEIS

%I #11 Apr 19 2022 11:41:40

%S 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 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 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 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 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 Antti Karttunen, <a href="/A353377/b353377.txt">Table of n, a(n) for n = 1..65537</a>

%F For all n >= 1, a(n) <= A353337(n).

%e 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 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 (PARI)

%o A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2));

%o 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 Cf. A001222, A007814, A345452, A353374, A353378 [= a(n^2)].

%Y Cf. also A353333, A353337, A353353, A353373.

%K nonn

%O 1,16

%A _Antti Karttunen_, Apr 17 2022