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
KEYWORD
sign,mult,easy
AUTHOR
Mats Granvik, May 18 2021
STATUS
approved