A369308 The number of square divisors d of n such that n/d is also a square.

1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Amiram Eldar, Jan 19 2024

Keywords

• nonn
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000037, A001620, A046951, A369307.

Programs

Mathematica

f[p_,e_] := If[OddQ[e], 0, e/2+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> if(x%2, 0, x/2 + 1), factor(n)[,2]));

Python

from math import prod
from sympy import factorint
def A369308(n): return prod(0 if e&1 else (e>>1)+1 for e in factorint(n).values()) # Chai Wah Wu, Jan 19 2024

Formula

Multiplicative with a(p^e) = 0 is e is odd, and e/2 + 1 if e is even.

a(n) = 0, if n is not a square (A000037), and A000005(sqrt(n)) otherwise.

Dirichlet g.f.: zeta(2*s)^2.

Sum_{k=1..n} a(k) ~ sqrt(n) * (log(n)/2 + 2*gamma - 1), where gamma is Euler's constant (A001620).