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A198302 a(n) = Sum_{d|n} d * sigma(n/d, d).

%I A198302 #19 Nov 05 2024 19:04:21
%S A198302 1,5,7,21,11,65,15,133,106,245,23,1077,27,1041,1637,3365,35,9992,39,
%T A198302 18361,16401,22841,47,134461,15686,106917,179494,355173,59,1220075,63,
%U A198302 1593189,1952705,2228909,631005,13778268,75,9962313,20732901,34805473,83,113693883
%N A198302 a(n) = Sum_{d|n} d * sigma(n/d, d).
%C A198302 Here sigma(n,k) is the sum of the k-th powers of the divisors of n.
%C A198302 Logarithmic derivative of A198301.
%H A198302 Robert Israel, <a href="/A198302/b198302.txt">Table of n, a(n) for n = 1..6236</a>
%F A198302 L.g.f.: Sum_{n>=1} Sum_{k>=1} sigma(n,k) * x^(n*k)/n.
%e A198302 L.g.f.: L(x) = x + 5*x^2/2 + 7*x^3/3 + 21*x^4/4 + 11*x^5/5 + 65*x^6/6 + ...
%e A198302 Exponentiation yields the g.f. of A198301: exp(L(x)) = 1 + x + 3*x^2 + 5*x^3 + 12*x^4 + 18*x^5 + 42*x^6 + 62*x^7 + 131*x^8 + 206*x^9 + 398*x^10 + ... + A198301(n)*x^n + ...
%p A198302 f:= proc(n) local d; add(d*numtheory:-sigma[d](n/d),d=numtheory:-divisors(n)) end proc:
%p A198302 map(f, [$1..100]); # _Robert Israel_, Nov 05 2024
%t A198302 a[n_] := DivisorSum[n, # * DivisorSigma[#, n/#] &]; Array[a, 40] (* _Amiram Eldar_, Aug 18 2023 *)
%o A198302 (PARI) {a(n)=sumdiv(n, d, d*sigma(n/d,d))}
%o A198302 (PARI) {a(n)=n*polcoeff(sum(m=1,n,sum(k=1,n\m,sigma(m,k)*x^(m*k)/m)+x*O(x^n)),n)}
%Y A198302 Cf. A000203, A198301 (exp), A185301, A198299.
%K A198302 nonn,look
%O A198302 1,2
%A A198302 _Paul D. Hanna_, Jan 27 2012