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A054598 a(0)=0; for n>0, a(n) = Sum_{d|n} d*2^(n/d).

%I A054598 #18 Dec 01 2018 11:02:01
%S A054598 0,2,8,14,32,42,104,142,320,554,1128,2070,4352,8218,16696,32934,66176,
%T A054598 131106,263480,524326,1050912,2097634,4198488,8388654,16786688,
%U A054598 33554642,67125352,134219390,268468960,536870970,1073811144,2147483710,4295099648,8589940890
%N A054598 a(0)=0; for n>0, a(n) = Sum_{d|n} d*2^(n/d).
%C A054598 Row sums of A322200, where A322200 describes Sum_{n>=1} -log(1 - (x^n + y^n)). - _Paul D. Hanna_, Dec 01 2018
%F A054598 L.g.f.: -log(Product_{ k>0 } (1-2*x^k)) = Sum_{ n>=0 } (a(n)/n)*x^n. - _Benedict W. J. Irwin_, Jun 23 2016
%F A054598 G.f.: Sum_{k>=1} 2^k*x^k/(1 - x^k)^2. - _Ilya Gutkovskiy_, Oct 24 2018
%t A054598 Table[CoefficientList[Series[-Log[-QPochhammer[2, x]], {x, 0, 60}], x][[n]] (n - 1), {n, 1, 60}] (* _Benedict W. J. Irwin_, Jun 23 2016 *)
%o A054598 (PARI) a(n) = sumdiv(n, d, d*2^(n/d)); \\ _Michel Marcus_, Jul 01 2016
%Y A054598 Cf. A054599, A054600, A054601, A000016, A000031.
%Y A054598 Cf. A322200.
%K A054598 nonn
%O A054598 0,2
%A A054598 _N. J. A. Sloane_, Apr 16 2000