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%I #22 Jan 17 2024 07:50:29
%S 1,-3,-7,-2,-21,21,-43,-4,-12,63,-111,14,-157,129,147,-8,-273,36,-343,
%T 42,301,333,-507,28,-80,471,-36,86,-813,-441,-931,-16,777,819,903,24,
%U -1333,1029,1099,84,-1641,-903,-1807,222,252,1521,-2163,56,-252,240,1911,314,-2757,108,2331
%N Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(s-2)).
%C Dirichlet inverse of A057660.
%H Amiram Eldar, <a href="/A328502/b328502.txt">Table of n, a(n) for n = 1..10000</a>
%H Olivier Bordelles, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Bordelles/bord21.html">A Multidimensional Cesaro Type Identity and Applications</a>, J. Int. Seq. 18 (2015) # 15.3.7.
%F a(1) = 1; a(n) = -Sum_{d|n, d<n} A057660(n/d) * a(d).
%F a(n) = Sum_{d|n} phi(n/d) * mu(d) * d^2.
%F 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
%F 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
%t 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}]
%t Table[DivisorSum[n, EulerPhi[n/#] MoebiusMu[#] #^2 &], {n, 1, 55}]
%t 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 *)
%o (PARI) a(n)={sumdiv(n, d, eulerphi(n/d)*moebius(d)*d^2)} \\ _Andrew Howroyd_, Oct 25 2019
%Y Cf. A000010, A008683, A030230 (positions of negative terms), A057660, A101035.
%K sign,mult
%O 1,2
%A _Ilya Gutkovskiy_, Oct 22 2019