A332794 a(n) = Sum_{d|n} (-1)^(d + 1) * d * phi(n/d).

1, -1, 5, -4, 9, -5, 13, -12, 21, -9, 21, -20, 25, -13, 45, -32, 33, -21, 37, -36, 65, -21, 45, -60, 65, -25, 81, -52, 57, -45, 61, -80, 105, -33, 117, -84, 73, -37, 125, -108, 81, -65, 85, -84, 189, -45, 93, -160, 133, -65, 165, -100, 105, -81, 189

Offset

1,3

Links

Table of n, a(n) for n=1..55.

Olivier Bordelles, A Multidimensional Cesaro Type Identity and Applications, J. Int. Seq. 18 (2015) # 15.3.7.

Formula

G.f.: Sum_{k>=1} phi(k) * x^k / (1 + x^k)^2.

Dirichlet g.f.: zeta(s-1)^2 * (1 - 2^(2 - s)) / zeta(s).

a(n) = Sum_{k=1..n} gcd(n, k) if n odd, Sum_{k=1..n} (-1)^(k + 1) * gcd(n, k) if n even.

From Amiram Eldar, Nov 04 2022: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*log(2)/Pi^2 = 0.210691... . (End)

a(2*n) = - Sum_{k = 1..n} gcd(2*k, n) = - A344372(n); a(2*n+1) = A018804(2*n+1). - Peter Bala, Jan 11 2024

a(n) = Sum_{k = 1..n} (-1)^(1 + gcd(k, n)) * gcd(k, n) (follows from an identity of Cesàro. See, for example, Bordelles, Lemma 1). - Peter Bala, Jan 16 2024

Mathematica

a[n_] := Sum[(-1)^(d + 1) d EulerPhi[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := If[OddQ[n], Sum[GCD[n, k], {k, 1, n}], Sum[(-1)^(k + 1) GCD[n, k], {k, 1, n}]]; Table[a[n], {n, 1, 55}]
f[p_, e_] := (e*(p-1) + p)*p^(e-1); f[2, e_] := -e*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, (-1)^(d+1)*d*eulerphi(n/d)); \\ Michel Marcus, Feb 24 2020

Crossrefs

Cf. A000010, A018804, A078747, A143520, A181983, A193356, A199084, A344372, A368929.

Sequence in context: A019776 A057763 A198609 * A054508 A359092 A110617

Adjacent sequences: A332791 A332792 A332793 * A332795 A332796 A332797

Keyword

sign,mult,easy

Author

Ilya Gutkovskiy, Feb 24 2020

Status

approved