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A353414 Number of ways to write n as a product of the terms of A353355 larger than 1; a(1) = 1 by convention (an empty product).

%I A353414 #11 Apr 19 2022 22:45:25
%S A353414 1,0,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0,1,1,1,1,1,0,0,0,2,1,0,
%T A353414 1,3,0,1,0,0,0,0,0,1,1,0,0,3,1,1,1,1,0,1,0,1,0,1,0,2,0,0,1,3,1,0,0,1,
%U A353414 1,0,0,4,0,1,1,1,1,0,0,1,1,0,0,2,0,1,0,0,0,2,0,1,1,0,1,3,0,1,1,2,0,0,0,1,0
%N A353414 Number of ways to write n as a product of the terms of A353355 larger than 1; a(1) = 1 by convention (an empty product).
%C A353414 Number of factorizations of n into factors k > 1 for which A353354(k) = 0.
%H A353414 Antti Karttunen, <a href="/A353414/b353414.txt">Table of n, a(n) for n = 1..65537</a>
%H A353414 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353414 a(p) = 0 for all primes p.
%F A353414 a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%e A353414 Of the fourteen divisors of 144 larger than 1, only [4, 6, 8, 9, 12, 18, 36, 48, 72, 144] are in A353355. Using combinations of these, 144 can be factored in six different ways as 144 = 36*4 = 18*8 = 12*12 = 9*4*4 = 6*6*4, therefore a(144) = 6.
%o A353414 (PARI)
%o A353414 A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };
%o A353414 A353354(n) = sumdiv(n,d,A332823(d));
%o A353414 A353380(n) = (0==A353354(n));
%o A353414 A353414(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353380(d), s += A353414(n/d, d))); (s));
%Y A353414 Cf. A003961, A048675, A332823, A348717, A353352, A353354, A353355, A353380, A353415 [= a(n^2)].
%Y A353414 Cf. also A353303, A353353.
%K A353414 nonn
%O A353414 1,32
%A A353414 _Antti Karttunen_, Apr 19 2022