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A078308 a(n) = Sum_{d divides n} d^(n/d + 1).

%I A078308 #27 Aug 02 2025 15:08:11
%S A078308 1,5,10,25,26,80,50,161,163,290,122,988,170,796,1580,2305,290,5561,
%T A078308 362,10670,9404,5912,530,58436,16251,19258,66340,118640,842,381740,
%U A078308 962,431105,547172,268214,509500,3534037,1370,1056880,4813052,8616326,1682
%N A078308 a(n) = Sum_{d divides n} d^(n/d + 1).
%H A078308 Seiichi Manyama, <a href="/A078308/b078308.txt">Table of n, a(n) for n = 1..6284</a>
%F A078308 G.f.: Sum_{n>0} n^2*x^n/(1-n*x^n).
%F A078308 L.g.f.: -log(Product_{ k>0 } (1-k*x^k)) = Sum_{ n>=0 } (a(n)/n)*x^n. - _Benedict W. J. Irwin_, Jul 04 2016
%p A078308 A078308 := proc(n)
%p A078308         add( d^(n/d+1),d=numtheory[divisors](n)) ;
%p A078308 end proc:
%p A078308 seq(A078308(n),n=1..10) ; # _R. J. Mathar_, Dec 14 2011
%t A078308 Table[CoefficientList[Series[-Log[Product[(1 - k x^k), {k, 1, 60}]], {x, 0, 60}],x][[n + 1]] (n), {n, 1, 60}] (* _Benedict W. J. Irwin_, Jul 04 2016 *)
%t A078308 Table[Total[#^(n/#+1)&/@Divisors[n]],{n,50}] (* _Harvey P. Dale_, Aug 02 2025 *)
%o A078308 (PARI) a(n) = sumdiv(n, d, d^(n/d+1)); \\ _Michel Marcus_, Jul 04 2016
%o A078308 (PARI) N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, 1-k*x^k)))) \\ _Seiichi Manyama_, Jun 02 2019
%Y A078308 Cf. A055225, A006906, A022661.
%K A078308 nonn
%O A078308 1,2
%A A078308 _Vladeta Jovovic_, Nov 22 2002