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

1,9

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

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

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]

Prog

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

Crossrefs

Cf. A002117, A013661, A016754, A034444, A056624, A298735.

Sequence in context: A031242 A340102 A031260 * A298735 A055090 A290106

Adjacent sequences: A360154 A360155 A360156 * A360158 A360159 A360160

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 29 2023

Status

approved