A353359 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).

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, 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, 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

Offset: 1

Author

Antti Karttunen, Apr 16 2022

Keywords

• nonn

Comments

Number of factorizations of n^3 into factors k > 1 for which A048675(k) is a multiple of three.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16415
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000578, A003961, A048675, A332820, A348717, A353350, A353353.

Cf. also A353304, A353334, A353378.

Programs

PARI

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A353350(n) = (0==(A048675(n)%3));
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));
A353359(n) = A353353(n^3);

Formula

a(n) = A353353(A000578(n)).

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

a(p) = 1 for all primes p.