A353334 Number of factorizations of the square of n into factors k > 1 for which both A001222(k) and A056239(k) are even.

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, 7, 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, 17, 1, 3, 6, 11, 2, 7, 1, 6, 2, 7, 1, 16, 1, 2, 4, 4, 2, 7, 1, 21, 5, 3, 1, 16, 3, 2, 3, 12, 1, 16, 3, 6, 2, 3, 2, 19, 1, 4, 4, 16, 1, 7

Offset: 1

Author

Antti Karttunen, Apr 14 2022

Keywords

• nonn

Comments

Number of factorizations of n^2 into terms of A340784 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, A001222, A003961, A056239, A340784, A348717, A353331, A353333.

Differs from A353304 for the first time at n=30, where a(30) = 7, while A353304(30) = 8.

Programs

PARI

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));
A353333(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (d<=m) && A353331(d), s += A353333(n/d, d))); (s));
A353334(n) = A353333(n^2);

Formula

a(n) = A353333(A000290(n)).

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

a(p) = 1 for all primes p.