A353304 Number of factorizations of n^2 into factors k > 1 for which A156552(k) is a multiple of three.

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 4, 1, 2, 2, 5, 1, 4, 1, 6, 3, 3, 1, 7, 2, 2, 3, 4, 1, 8, 1, 7, 2, 3, 2, 9, 1, 2, 3, 12, 1, 7, 1, 6, 4, 3, 1, 12, 2, 6, 2, 4, 1, 7, 3, 7, 3, 2, 1, 19, 1, 3, 6, 11, 2, 8, 1, 6, 2, 7, 1, 16, 1, 2, 4, 4, 2, 7, 1, 21, 5, 3, 1, 16, 3, 2, 3, 12, 1, 18, 3, 6, 2, 3, 2, 19, 1, 4, 4, 16, 1, 8

Offset: 1

Author

Antti Karttunen, Apr 10 2022

Keywords

• nonn

Comments

Number of factorizations of the square of n into terms of A329609 that are larger than one.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10080
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000290, A003961, A156552, A329609, A348717, A353269, A353303.

Programs

PARI

A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A353269(n) = (!(A156552(n)%3));
A353303(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&
A353269(d), s += A353303(n/d, d))); (s));
A353304(n) = A353303(n^2);

Formula

a(n) = A353303(A000290(n)).

a(p) = 1 for all primes p.

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