A353378 - OEIS

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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

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OFFSET

1,4

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

FORMULA

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

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

PROG

(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);

CROSSREFS

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