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A351654 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(s-2)).

%I A351654 #15 May 26 2025 09:54:34
%S A351654 1,-5,-11,3,-29,55,-55,3,16,145,-131,-33,-181,275,319,3,-305,-80,-379,
%T A351654 -87,605,655,-551,-33,96,905,16,-165,-869,-1595,-991,3,1441,1525,1595,
%U A351654 48,-1405,1895,1991,-87,-1721,-3025,-1891,-393,-464,2755,-2255,-33,288,-480,3355,-543,-2861,-80,3799
%N A351654 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(s-2)).
%C A351654 Dirichlet inverse of A069097.
%H A351654 Amiram Eldar, <a href="/A351654/b351654.txt">Table of n, a(n) for n = 1..10000</a>
%F A351654 a(1) = 1; a(n) = -Sum_{d|n, d < n} A069097(n/d) * a(d).
%F A351654 a(n) = Sum_{d|n} A023900(n/d) * A334657(d).
%F A351654 a(n) = Sum_{d|n} A046970(n/d) * A055615(d).
%F A351654 a(n) = Sum_{d|n} A000005(n/d) * A328254(d).
%F A351654 Multiplicative with a(p) = -p^2 - p + 1, and a(p^e) = p^3 - p^2 - p + 1 for e >= 2. - _Amiram Eldar_, May 25 2025
%t A351654 A069097[n_] := Sum[GCD[n, k]^2, {k, 1, n}]; a[1] = 1; a[n_] := a[n] = -Sum[A069097[n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
%t A351654 f[p_, e_] := If[e == 1, 0, p^3] - p^2 - p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 25 2025 *)
%o A351654 (PARI)
%o A351654 up_to = 20000;
%o A351654 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A351654 A069097(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d));
%o A351654 v351654 = DirInverseCorrect(vector(up_to, n, A069097(n)));
%o A351654 A351654(n) = v351654[n]; \\ _Antti Karttunen_, Feb 16 2022
%o A351654 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)*(1 - p^2*X)/(1 - X))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 16 2022
%Y A351654 Cf. A000005, A023900, A046970, A055615, A069097, A101035, A328254, A334657.
%K A351654 sign,mult
%O A351654 1,2
%A A351654 _Ilya Gutkovskiy_, Feb 16 2022