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

1,4

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.

Cf. also A353304, A353334, A353338, A353359.

Sequence in context: A277692 A354992 A129138 * A112970 A112971 A299201

Adjacent sequences: A353375 A353376 A353377 * A353379 A353380 A353381

Keyword

nonn

Author

Antti Karttunen, Apr 17 2022

Status

approved