A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)*x^n = log( Sum_{n>=0} A107742(n)*x^n ).

1, 3, 7, 7, 11, 21, 15, 15, 34, 33, 23, 49, 27, 45, 77, 31, 35, 102, 39, 77, 105, 69, 47, 105, 86, 81, 142, 105, 59, 231, 63, 63, 161, 105, 165, 238, 75, 117, 189, 165, 83, 315, 87, 161, 374, 141, 95, 217, 162, 258, 245, 189, 107, 426, 253, 225, 273, 177, 119, 539, 123, 189, 510, 127, 297

Offset: 1

Author

Paul D. Hanna, Jun 26 2005

Keywords

• nonn
• easy
• mult

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001227, A060640, A107742.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), this sequence (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).

Programs

Mathematica

a[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2]&]&]; Array[a, 65] (* Jean-François Alcover, Dec 23 2015 *)
f[p_, e_] := ((p + e*(p-1) - 2)*p^(e+1) + 1)/(p-1)^2; f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

PARI

a(n)=sumdiv(n,d,d*sumdiv(d,m,m%2))

PARI

N=66; x='x+O('x^N); /* that many terms */
c=sum(j=1, N, j*x^j);
t=log( 1/prod(j=0, N, eta(x^(2*j+1))) );
gf=serconvol(t, c);
Vec(gf) /* show terms */
/* Joerg Arndt, May 03 2008 */

Formula

a(n) = Sum_{d|n} d * Sum_{m|d} (m mod 2).

G.f.: Sum_{n>=1} a(n)/n*x^n = Sum_{j>=1} Sum_{i>=1} log(1+x^(i*j)).

From Vladeta Jovovic, Jul 05 2005:(Start)

Multiplicative with a(2^e) = 2^(e+1)-1 and a(p^e) = (p^(e+2)*(e+1)-p^(e+1)*(e+2)+1)/(p-1)^2 for p>2.

G.f.: Sum_{n>0} n*A000005(n)*x^n/(1+x^n).

G.f.: Sum_{n>0} n*A001227(n)*x^n/(1-x^n).

a(n) = A060640(n) if n is odd, else a(n) = A060640(n) - 2*A060640(n/2).

a(n) = Sum_{d|n} d*A001227(d).

a(n) = Sum_{d|n} d*A000593(n/d).

A107742(n) = (1/n)*Sum_{k=1..n} a(k)*A107742(n-k). (End)