%I #23 Apr 15 2022 10:33:20

%S 1,0,0,1,0,0,0,0,1,1,0,0,0,0,0,2,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,1,

%T 0,2,0,0,1,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,1,0,3,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,2,1,0,1,2,0,2,1,0,0,1,0,0,0,0,0,3,0,0,0,0,0

%N Number of ways to write n as a product of the terms of A340784 larger than 1; a(1) = 1 by convention (an empty product).

%C Number of factorizations of n into factors k > 1 for which both A001222(k) and A056239(k) are even.

%H Antti Karttunen, <a href="/A353333/b353333.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(p) = 0 for all primes p.

%F a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

%e Of the eleven divisors of 220 larger than one, only [4, 10, 22, 55, 220] are in A340784, as both the number of their prime factors (with repetition, A001222), [2, 2, 2, 2, 4], and their integer pseudo logarithms (A056239), [2, 4, 6, 8, 10], are even. Using these factors gives the following possible factorizations: 220 = 22*10 = 55*4, therefore a(220) = 3.

%e Of the eight divisors of 256 larger than one, only [1, 4, 16, 64, 256] are in A340784. Using these factors, we obtain the following factorizations: 256 = 64*4 = 16*16 = 16*4*4 = 4*4*4*4, therefore a(256) = 5.

%e Of the 23 divisors of 792 larger than one, only [4, 9, 22, 36, 88, 198, 792] are in A340784. Using these factors gives the following possible factorizations: 792 = 198*4 = 88*9 = 36*22 = 22*4*9, therefore a(792) = 5.

%o (PARI)

%o A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }

%o A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));

%o A353333(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (d<=m) && A353331(d), s += A353333(n/d, d))); (s));

%Y Cf. A001222, A003961, A056239, A340784, A348717, A353331, A353334 [= a(n^2)].

%Y Differs from A353303 for the first time at n=30, where a(30) = 0, while A353303(30) = 1.

%K nonn

%O 1,16

%A _Antti Karttunen_, Apr 14 2022