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Dirichlet g.f.: zeta(s)^2/zeta(2*s-1).
3

%I #42 Dec 25 2022 03:56:41

%S 1,2,2,1,2,4,2,0,0,4,2,2,2,4,4,-1,2,0,2,2,4,4,2,0,-2,4,-2,2,2,8,2,-2,

%T 4,4,4,0,2,4,4,0,2,8,2,2,0,4,2,-2,-4,-4,4,2,2,-4,4,0,4,4,2,4,2,4,0,-3,

%U 4,8,2,2,4,8,2,0,2,4,-4,2,4,8,2,-2,-4,4,2

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

%C Compare with the Dirichlet generating function for the matrix in A191898.

%H Antti Karttunen, <a href="/A344326/b344326.txt">Table of n, a(n) for n = 1..65537</a>

%F Dirichlet g.f.: zeta(s)^2/zeta(2*s-1). After _Franklin T. Adams-Watters_ in A034448.

%F Multiplicative with a(p^e) = 2 - (e-1)*(p-1). - _Amiram Eldar_, May 18 2021

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - _Amiram Eldar_, Nov 18 2022

%t nn = 83; usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; U=Table[Table[If[Mod[n, k] == 0, usigma[n/k], 0], {k, 1, nn}], {n, 1, nn}]; M=Table[Table[If[Mod[n, k] == 0, MoebiusMu[n/k]*n/k, 0], {k, 1, nn}], {n, 1, nn}]; Z=Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; (U.M.Z)[[Range[nn], 1]] (* After Giovanni Resta A034448 *)

%t f[p_, e_] := 2 - (e-1)*(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 18 2021 *)

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X^2)/(1-X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, May 18 2021

%o (PARI) A344326(n) = if(1==n,1, my(f=factor(n)); prod(k=1,#f~,(2-((f[k,1]-1)*(f[k,2]-1))))); \\ (After the multiplicative formula given by _Amiram Eldar_) - _Antti Karttunen_, May 19 2021

%Y Cf. A034444, A034448, A191898.

%K sign,mult,easy

%O 1,2

%A _Mats Granvik_, May 18 2021