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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 ).

%I A109386 #34 Jun 29 2025 22:18:14
%S A109386 1,3,7,7,11,21,15,15,34,33,23,49,27,45,77,31,35,102,39,77,105,69,47,
%T A109386 105,86,81,142,105,59,231,63,63,161,105,165,238,75,117,189,165,83,315,
%U A109386 87,161,374,141,95,217,162,258,245,189,107,426,253,225,273,177,119,539,123,189,510,127,297
%N 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 ).
%H A109386 Seiichi Manyama, <a href="/A109386/b109386.txt">Table of n, a(n) for n = 1..10000</a>
%F A109386 a(n) = Sum_{d|n} d * Sum_{m|d} (m mod 2).
%F A109386 G.f.: Sum_{n>=1} a(n)/n*x^n = Sum_{j>=1} Sum_{i>=1} log(1+x^(i*j)).
%F A109386 From _Vladeta Jovovic_, Jul 05 2005:(Start)
%F A109386 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.
%F A109386 G.f.: Sum_{n>0} n*A000005(n)*x^n/(1+x^n).
%F A109386 G.f.: Sum_{n>0} n*A001227(n)*x^n/(1-x^n).
%F A109386 a(n) = A060640(n) if n is odd, else a(n) = A060640(n) - 2*A060640(n/2).
%F A109386 a(n) = Sum_{d|n} d*A001227(d).
%F A109386 a(n) = Sum_{d|n} d*A000593(n/d).
%F A109386 A107742(n) = (1/n)*Sum_{k=1..n} a(k)*A107742(n-k). (End)
%t A109386 a[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2]&]&]; Array[a, 65] (* _Jean-François Alcover_, Dec 23 2015 *)
%t A109386 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 *)
%o A109386 (PARI) a(n)=sumdiv(n,d,d*sumdiv(d,m,m%2))
%o A109386 (PARI)N=66; x='x+O('x^N); /* that many terms */
%o A109386 c=sum(j=1, N, j*x^j);
%o A109386 t=log( 1/prod(j=0, N, eta(x^(2*j+1))) );
%o A109386 gf=serconvol(t, c);
%o A109386 Vec(gf) /* show terms */
%o A109386 /* _Joerg Arndt_, May 03 2008 */
%Y A109386 Cf. A000005, A001227, A060640, A107742.
%Y A109386 Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), this sequence (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).
%K A109386 nonn,easy,mult
%O A109386 1,2
%A A109386 _Paul D. Hanna_, Jun 26 2005