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%I #8 Apr 17 2022 09:38:12
%S 1,1,1,2,1,4,1,3,2,3,1,9,1,4,4,5,1,9,1,8,3,3,1,19,2,4,3,9,1,26,1,7,4,
%T 3,4,29,1,4,3,16,1,26,1,8,9,3,1,36,2,8,4,9,1,19,3,19,3,4,1,89,1,3,8,
%U 11,4,26,1,8,4,26,1,67,1,4,9,9,4,26,1,31,5,3,1,91,3,4,3,16,1,91,3,8,4,3,4,64,1,9,9
%N Number of ways to write the cube of n as a product of the terms of A332820 larger than 1; a(1) = 1 by convention (an empty product).
%C Number of factorizations of n^3 into factors k > 1 for which A048675(k) is a multiple of three.
%H Antti Karttunen, <a href="/A353359/b353359.txt">Table of n, a(n) for n = 1..16415</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F a(n) = A353353(A000578(n)).
%F a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%F a(p) = 1 for all primes p.
%o (PARI)
%o A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A353350(n) = (0==(A048675(n)%3));
%o A353353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353350(d), s += A353353(n/d, d))); (s));
%o A353359(n) = A353353(n^3);
%Y Cf. A000578, A003961, A048675, A332820, A348717, A353350, A353353.
%Y Cf. also A353304, A353334, A353378.
%K nonn
%O 1,4
%A _Antti Karttunen_, Apr 16 2022