OFFSET
1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
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) = Sum_{d|n} d*A001227(d).
a(n) = Sum_{d|n} d*A000593(n/d).
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Paul D. Hanna, Jun 26 2005
STATUS
approved