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

1,36

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Formula

a(p) = 0 for all primes p.

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

Example

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.

Prog

(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));

Crossrefs

Cf. A031215, A031368, A325698, A353370.

Cf. also A353333, A353353, A353377.

Sequence in context: A138954 A245811 A258822 * A064530 A307832 A037047

Adjacent sequences: A353370 A353371 A353372 * A353374 A353375 A353376

Keyword

nonn

Author

Antti Karttunen, Apr 16 2022

Status

approved