A078439 a(n) = Sum_{k=1..n} gcd(k,n)*mu(gcd(k,n))^2.

1, 3, 5, 4, 9, 15, 13, 8, 12, 27, 21, 20, 25, 39, 45, 16, 33, 36, 37, 36, 65, 63, 45, 40, 40, 75, 36, 52, 57, 135, 61, 32, 105, 99, 117, 48, 73, 111, 125, 72, 81, 195, 85, 84, 108, 135, 93, 80, 84, 120, 165, 100, 105, 108, 189, 104, 185, 171, 117, 180, 121, 183, 156, 64

Offset

1,2

Links

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

Formula

a(n) = Sum_{d|n} d*mu(d)^2*phi(n/d).

Multiplicative with a(p) = 2*p-1 and a(p^e) = 2*(p-1)*p^(e-1), e>1.

Dirichlet g.f.: zeta(s-1)^2 / (zeta(s) * zeta(2s-2)). - Álvar Ibeas, Mar 20 2015

Sum_{k=1..n} a(k) ~ 9 * n^2 * (2*log(n) + 4*gamma - 1 - 36*Zeta'(2)/Pi^2) / Pi^4, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 01 2019

Mathematica

Table[Sum[d*MoebiusMu[d]^2*EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Feb 01 2019 *)
f[p_, e_] := If[e==1, 2*p-1, 2*(p-1)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

Prog

(PARI) vector(80, n, sumdiv(n, d, d*moebius(d)^2*eulerphi(n/d))) \\ Michel Marcus, Mar 20 2015
(Magma) [&+[Gcd(k, n)*MoebiusMu(Gcd(n, k))^2:k  in [1..n]]:n in [1..70]]; // Marius A. Burtea, Sep 15 2019

Crossrefs

Cf. A001620, A000010, A008683, A063659.

Sequence in context: A177983 A294673 A347128 * A007063 A335500 A127397

Adjacent sequences: A078436 A078437 A078438 * A078440 A078441 A078442

Keyword

mult,nonn

Author

Vladeta Jovovic, Dec 31 2002

Status

approved