A319998 a(n) = Sum_{d|n, d is even} mu(n/d)*d, where mu(n) is Moebius function A008683.

0, 2, 0, 2, 0, 4, 0, 4, 0, 8, 0, 4, 0, 12, 0, 8, 0, 12, 0, 8, 0, 20, 0, 8, 0, 24, 0, 12, 0, 16, 0, 16, 0, 32, 0, 12, 0, 36, 0, 16, 0, 24, 0, 20, 0, 44, 0, 16, 0, 40, 0, 24, 0, 36, 0, 24, 0, 56, 0, 16, 0, 60, 0, 32, 0, 40, 0, 32, 0, 48, 0, 24, 0, 72, 0, 36, 0, 48, 0, 32, 0, 80, 0, 24, 0, 84, 0, 40, 0, 48, 0, 44, 0, 92, 0, 32, 0, 84, 0, 40, 0, 64, 0, 48, 0

Offset: 1

Author

Antti Karttunen, Oct 31 2018

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000010, A008683, A059841, A140434, A146076, A319997.

Programs

Mathematica

Rest[CoefficientList[Series[Sum[2*MoebiusMu[k]*x^(2*k)/(1 - x^(2*k))^2, {k, 1, 100}], {x, 0, 100}], x]] (* Vaclav Kotesovec, Nov 03 2018 *)

PARI

A319998(n) = sumdiv(n,d,(!(d%2))*moebius(n/d)*d);

PARI

A319998(n) = if(n%2, 0, 2*eulerphi(n/2));

Formula

a(n) = Sum_{d|n} A059841(d)*A008683(n/d)*d.

a(n) = A000010(n) - A319997(n).

a(2n) = 2*A000010(n), a(2n+1) = 0.

G.f.: Sum_{k>=1} 2*mu(k)*x^(2*k)/(1 - x^(2*k))^2. - Ilya Gutkovskiy, Nov 02 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/(2*Pi^2) = 0.151981... . - Amiram Eldar, Nov 12 2022