A035109 Numerators in the expansion of the Dirichlet series zeta(s) * Product((1+p^-s) / (1-p^(1-s))), p > 2.
1, 1, 5, 1, 7, 5, 9, 1, 17, 7, 13, 5, 15, 9, 35, 1, 19, 17, 21, 7, 45, 13, 25, 5, 37, 15, 53, 9, 31, 35, 33, 1, 65, 19, 63, 17, 39, 21, 75, 7, 43, 45, 45, 13, 119, 25, 49, 5, 65, 37, 95, 15, 55, 53, 91, 9, 105, 31, 61, 35, 63, 33, 153, 1, 105, 65, 69, 19, 125, 63, 73, 17, 75, 39
Offset: 0
Examples
a(6) = (1/6)*(mu(6)*1*1 + mu(3)*3*1 + mu(2)*4*4 + mu(1)*12*4) = 5. - _Thomas Ward_, Apr 08 2009
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- M. Baake and R. V. Moody, Similarity submodules and semigroups in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.
- A. Pakapongpun and T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 17.
Programs
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Mathematica
a[n_] := (1/n)*DivisorSum[n, MoebiusMu[n/#]*DivisorSigma[1, #]*DivisorSum[ #, If[OddQ[#], #, 0]&]&]; Array[a, 80] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)
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PARI
a(n)=(1/n)*sumdiv(n,d,moebius(n/d)*sigma(d)*sumdiv(d,e,if(e%2,e,0))) \\ Thomas Ward, Apr 08 2009
Formula
Dirichlet g.f.: zeta(s) * Product((1+p^-s) / (1-p^(1-s))), p > 2.
a(n) = (1/n) * Sum_{d|n} mu(n/d) * (Sum_{e|d} e) * (Sum_{e|d, e odd only} e). - Thomas Ward, Apr 08 2009
From Ridouane Oudra, Jun 18 2025: (Start)
a(n) = Sum_{d|n} (psi(2*d) - 2*psi(d)), where psi = A001615.
a(n) = Sum_{d|n, d odd} psi(d).
a(n) = A309324(n) / gcd(n,2).
a(2*n) = a(n).
a(2*n+1) = A060648(2*n+1). (End)
From Vaclav Kotesovec, Jun 21 2025: (Start)
Dirichlet g.f.: (1 - 2^(1-s)) * zeta(s-1) * zeta(s)^2 / ((1 + 2^(-s)) * zeta(2*s)).
Sum_{k=1..n} a(k) ~ n^2/2. (End)
Comments