A344326 Dirichlet g.f.: zeta(s)^2/zeta(2*s-1).

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, 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, 4, 8, 2, 2, 4, 8, 2, 0, 2, 4, -4, 2, 4, 8, 2, -2, -4, 4, 2

Offset

1,2

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

Dirichlet g.f.: zeta(s)^2/zeta(2*s-1). After Franklin T. Adams-Watters in A034448.

Multiplicative with a(p^e) = 2 - (e-1)*(p-1). - Amiram Eldar, May 18 2021

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

Mathematica

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 *)
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 *)

Prog

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

Crossrefs

Cf. A034444, A034448, A191898.

Sequence in context: A127677 A238009 A231145 * A007427 A048106 A304649

Adjacent sequences: A344323 A344324 A344325 * A344327 A344328 A344329

Keyword

sign,mult,easy

Author

Mats Granvik, May 18 2021

Status

approved