cp's OEIS Frontend

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).

%I A353373 #15 Apr 27 2022 11:54:49
%S A353373 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,
%T A353373 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,
%U A353373 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
%N 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).
%C A353373 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.
%H A353373 Antti Karttunen, <a href="/A353373/b353373.txt">Table of n, a(n) for n = 1..65537</a>
%H A353373 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353373 a(p) = 0 for all primes p.
%F A353373 a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%e A353373 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.
%e A353373 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.
%o A353373 (PARI)
%o A353373 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
%o A353373 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));
%Y A353373 Cf. A031215, A031368, A325698, A353370.
%Y A353373 Cf. also A353333, A353353, A353377.
%K A353373 nonn
%O A353373 1,36
%A A353373 _Antti Karttunen_, Apr 16 2022