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%I #30 Apr 30 2023 02:06:57
%S 1,1,5,2,9,5,13,4,21,9,21,10,25,13,45,8,33,21,37,18,65,21,45,20,65,25,
%T 81,26,57,45,61,16,105,33,117,42,73,37,125,36,81,65,85,42,189,45,93,
%U 40,133,65,165,50,105,81,189,52,185,57,117,90
%N The sum of the odd terms in row n of A050873.
%H Amiram Eldar, <a href="/A255655/b255655.txt">Table of n, a(n) for n = 1..10000</a>
%F For odd n, a(n) = A018804(n).
%F Dirichlet g.f.: zeta(s-1)^2*(1 - 2^(1-s))/zeta(s).
%F a(n) = Sum_{d|n} A193356(d)*A000010(n/d). - _Werner Schulte_, Feb 04 2018
%F Multiplicative with a(2^e)=2^(e-1) for e>0 and a(p^e)=((p-1)*e+p)*p^(e-1) for e>0 and p>2. - _Werner Schulte_, Feb 04 2018
%F Sum_{k=1..n} a(k) ~ 3*n^2 / (2*Pi^2) * (log(n) - 1/2 + 2*gamma + log(2) - 6*Zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Feb 02 2019
%e a(10)=9 because row 10 of A050873 is gcd(10,k) for k=1,2,...10: 1, 2, 1, 2, 5, 2, 1, 2, 1, 10. If we sum the odd terms in this row we have 1+1+5+1+1=9.
%t nn = 60; f[list_, i_] := list[[i]]; a =Table[EulerPhi[n], {n, 1, nn}]; b =
%t Table[If[OddQ[n], n, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}]
%t Table[Sum[(d*(1-(-1)^d)/2)*EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 50}] (* _Vaclav Kotesovec_, Feb 02 2019 *)
%t f[p_, e_] := p^(e-1) * If[p == 2, 1, (p-1)*e + p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 30 2023 *)
%o (PARI) a(n) = sum(k=1, n, my(g = gcd(n, k)); if (g % 2, g, 0)); \\ _Michel Marcus_, Feb 05 2018
%Y Cf. A000010, A001620, A018804, A050873, A193356.
%K nonn,mult
%O 1,3
%A _Geoffrey Critzer_, Mar 01 2015
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