A353373 Number of ways to write n as a product of the terms of A325698 larger than 1; a(1) = 1 by convention (an empty product).

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Author

Antti Karttunen, Apr 16 2022

Keywords

• nonn

Comments

Number of factorizations of n into factors k > 1 for which there are equal number of odd-indexed primes (A031368) as there are even-indexed primes (A031215) in their prime factorization.

Examples

			Of the 23 divisors of 936 larger than 1, only [6, 26, 36, 156, 936] are in A325698. These can be combined to yield a product yielding 936 in four possible ways as 936 = 156*6 = 36*26 = 26*6*6, therefore a(936) = 4.
Of the 31 divisors of 2184 larger than 1, only [6, 14, 26, 84, 156, 364, 2184] are in A325698. These can be combined to yield a product yielding 2184 in five possible ways as 2184 = 364*6 = 156*14 = 84*26, therefore a(2184) = 5.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A031215, A031368, A325698, A353370.

Cf. also A353333, A353353, A353377.

Programs

PARI

A353370(n) = { my(f = factor(n)); (0==sum(i=1, #f~, f[i,2]*((-1)^(primepi(f[i, 1])%2)))); }; \\ Antti Karttunen, Apr 16 2022
A353373(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353370(d), s += A353373(n/d, d))); (s));

Formula

a(p) = 0 for all primes p.

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