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A035109 Numerators in the expansion of the Dirichlet series zeta(s) * Product((1+p^-s) / (1-p^(1-s))), p > 2.

%I A035109 #37 Jun 25 2025 01:01:32
%S A035109 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,
%T A035109 31,35,33,1,65,19,63,17,39,21,75,7,43,45,45,13,119,25,49,5,65,37,95,
%U A035109 15,55,53,91,9,105,31,61,35,63,33,153,1,105,65,69,19,125,63,73,17,75,39
%N A035109 Numerators in the expansion of the Dirichlet series zeta(s) * Product((1+p^-s) / (1-p^(1-s))), p > 2.
%C A035109 a(n) is also the number of orbits of length n for the map SxT where S has one orbit of each length and T has one orbit of each odd length. - _Thomas Ward_, Apr 08 2009
%H A035109 Vincenzo Librandi, <a href="/A035109/b035109.txt">Table of n, a(n) for n = 0..1000</a>
%H A035109 M. Baake and R. V. Moody, <a href="http://www.math.uni-bielefeld.de/baake/ps/fields3.ps.gz">Similarity submodules and semigroups</a> in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.
%H A035109 A. Pakapongpun and T. Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Ward/ward17.html">Functorial Orbit counting</a>, JIS 12 (2009) 09.2.4, example 17.
%F A035109 Dirichlet g.f.: zeta(s) * Product((1+p^-s) / (1-p^(1-s))), p > 2.
%F A035109 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
%F A035109 From _Ridouane Oudra_, Jun 18 2025: (Start)
%F A035109 a(n) = (1/n) * Sum_{d|n} mu(n/d) * A000203(d) * A000593(d).
%F A035109 a(n) = Sum_{d|n} (psi(2*d) - 2*psi(d)), where psi = A001615.
%F A035109 a(n) = Sum_{d|n, d odd} psi(d).
%F A035109 a(n) = A309324(n) / gcd(n,2).
%F A035109 a(n) = A309324(A000265(n)).
%F A035109 a(n) = A060648(A000265(n)).
%F A035109 a(2*n) = a(n).
%F A035109 a(2*n+1) = A060648(2*n+1). (End)
%F A035109 From _Vaclav Kotesovec_, Jun 21 2025: (Start)
%F A035109 Dirichlet g.f.: (1 - 2^(1-s)) * zeta(s-1) * zeta(s)^2 / ((1 + 2^(-s)) * zeta(2*s)).
%F A035109 Sum_{k=1..n} a(k) ~ n^2/2. (End)
%e A035109 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
%t A035109 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 *)
%o A035109 (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
%Y A035109 Cf. A008683, A000203, A000593, A001615, A309324, A060648, A000265.
%K A035109 nonn,easy
%O A035109 0,3
%A A035109 _N. J. A. Sloane_