A328502 Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(s-2)).

1, -3, -7, -2, -21, 21, -43, -4, -12, 63, -111, 14, -157, 129, 147, -8, -273, 36, -343, 42, 301, 333, -507, 28, -80, 471, -36, 86, -813, -441, -931, -16, 777, 819, 903, 24, -1333, 1029, 1099, 84, -1641, -903, -1807, 222, 252, 1521, -2163, 56, -252, 240, 1911, 314, -2757, 108, 2331

Offset

1,2

Comments

Dirichlet inverse of A057660.

Links

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

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

Formula

a(1) = 1; a(n) = -Sum_{d|n, d<n} A057660(n/d) * a(d).

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

Multiplicative with a(p) = p - 1 - p^2, and a(p^e) = -p^(e-1) * (p-1)^2, for e > 1. - Amiram Eldar, Dec 03 2022

a(n) = Sum_{k = 1..n} gcd(k, n)^2 * mu(gcd(k, n)) (follows fom an identity of Cesàro. See, for example, Bordelles, Lemma 1). - Peter Bala, Jan 16 2024

Mathematica

a[1] = 1; a[n_] := -Sum[DivisorSigma[2, (n/d)^2]/DivisorSigma[1, (n/d)^2] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
Table[DivisorSum[n, EulerPhi[n/#] MoebiusMu[#] #^2 &], {n, 1, 55}]
f[p_, e_] := If[e == 1, p - 1 - p^2, -p^(e - 1)*(p - 1)^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

Prog

(PARI) a(n)={sumdiv(n, d, eulerphi(n/d)*moebius(d)*d^2)} \\ Andrew Howroyd, Oct 25 2019

Crossrefs

Cf. A000010, A008683, A030230 (positions of negative terms), A057660, A101035.

Sequence in context: A245611 A063041 A257729 * A308480 A347772 A248214

Adjacent sequences: A328499 A328500 A328501 * A328503 A328504 A328505

Keyword

sign,mult

Author

Ilya Gutkovskiy, Oct 22 2019

Status

approved