A360157 a(n) is the number of unitary divisors of n that are odd squares.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 29 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A298735 at n = 27.

The unitary analog of A298735.

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, A016754, A034444, A056624, A298735.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], 1, 2]; f[2, e_] := 1; 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, 1, 2)));}

Formula

Multiplicative with a(2^e) = 1, 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)/(4^s + 2^s + 1).

Sum_{k=1..n} a(k) ~ c * n, where c = Pi^2/(7*zeta(3)) = 1.172942380817... .

More precise asymptotics: Sum_{k=1..n} a(k) ~ Pi^2 * n / (7*zeta(3)) + (4 + sqrt(2)) * zeta(1/2) * sqrt(n) / (7*zeta(3/2)). - Vaclav Kotesovec, Jan 29 2023