A353378 Number of ways to write the square of n as a product of the terms of A345452 larger than 1; a(1) = 1 by convention (an empty product).

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 3, 5, 1, 4, 1, 4, 3, 2, 1, 7, 2, 2, 3, 4, 1, 7, 1, 7, 3, 2, 3, 9, 1, 2, 3, 7, 1, 7, 1, 4, 6, 2, 1, 12, 2, 4, 3, 4, 1, 7, 3, 7, 3, 2, 1, 16, 1, 2, 6, 11, 3, 7, 1, 4, 3, 7, 1, 16, 1, 2, 6, 4, 3, 7, 1, 12, 5, 2, 1, 16, 3, 2, 3, 7, 1, 17, 3, 4, 3, 2, 3, 19, 1, 4, 6, 9, 1, 7

Offset: 1

Author

Antti Karttunen, Apr 17 2022

Keywords

• nonn

Comments

Number of factorizations of n^2 into factors k > 1 for which there is an even number of primes (when counted with multiplicity, A001222) in their prime factorization, and the 2-adic valuation of k (A007814) is also even.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000290, A001222, A007814, A345452, A353374, A353377.

Cf. also A353304, A353334, A353338, A353359.

Programs

PARI

A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2));
A353377(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353374(d), s += A353377(n/d, d))); (s));
A353378(n) = A353377(n^2);

Formula

a(n) = A353377(A000290(n)).

For all n >= 1, a(n) <= A353338(n).