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

1,2

Links

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

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

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 *)

Prog

(PARI) A319998(n) = sumdiv(n, d, (!(d%2))*moebius(n/d)*d);
(PARI) A319998(n) = if(n%2, 0, 2*eulerphi(n/2));

Crossrefs

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

Sequence in context: A362208 A167156 A132952 * A336120 A029187 A201863

Adjacent sequences: A319995 A319996 A319997 * A319999 A320000 A320001

Keyword

nonn

Author

Antti Karttunen, Oct 31 2018

Status

approved