A360158 a(n) is the number of unitary divisors of n that are odd squares minus the number of unitary divisors of n that are even squares.

1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 29 2023

Keywords

• nonn
• easy
• mult

Comments

The unitary analog of A344299.

The least term that is larger than 2 is a(225) = 4.

Links

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

Crossrefs

Cf. A002117, A013661, A016742, A016754, A056624, A344299, A350388, A360157.

Programs

Mathematica

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

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, 1, if(f[i, 1] == 2, 0, 2)));}

Formula

a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} (-1)^(d+1).

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

Dirichlet g.f.: (zeta(s)*zeta(2*s)/zeta(3*s)) * (4^s + 2^s - 1)/(4^s + 2^s + 1).

Sum_{k=1..n} a(k) ~ c * n, where c = 5*zeta(2)/(7*zeta(3)) = 0.977451984014... .