A353378 - OEIS

%I #10 Apr 19 2022 11:41:44

%S 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,

%T 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,

%U 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

%N 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).

%C 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.

%H Antti Karttunen, <a href="/A353378/b353378.txt">Table of n, a(n) for n = 1..65537</a>

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

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

%o (PARI)

%o A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2));

%o 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));

%o A353378(n) = A353377(n^2);

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

%Y Cf. also A353304, A353334, A353338, A353359.

%K nonn

%O 1,4

%A _Antti Karttunen_, Apr 17 2022