A255655 The sum of the odd terms in row n of A050873.

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, 81, 26, 57, 45, 61, 16, 105, 33, 117, 42, 73, 37, 125, 36, 81, 65, 85, 42, 189, 45, 93, 40, 133, 65, 165, 50, 105, 81, 189, 52, 185, 57, 117, 90

Offset: 1

Author

Geoffrey Critzer, Mar 01 2015

Keywords

• nonn
• mult

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A001620, A018804, A050873, A193356.

Programs

Mathematica

nn = 60; f[list_, i_] := list[[i]]; a =Table[EulerPhi[n], {n, 1, nn}]; b =
Table[If[OddQ[n], n, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}]
Table[Sum[(d*(1-(-1)^d)/2)*EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Feb 02 2019 *)
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 *)

PARI

a(n) = sum(k=1, n, my(g = gcd(n, k)); if (g % 2, g, 0)); \\ Michel Marcus, Feb 05 2018

Formula

For odd n, a(n) = A018804(n).

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

a(n) = Sum_{d|n} A193356(d)*A000010(n/d). - Werner Schulte, Feb 04 2018

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

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