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%I #12 Aug 20 2023 10:50:22
%S 1,5,7,21,11,65,15,133,106,245,23,1077,27,1041,1637,3365,35,9992,39,
%T 18361,16401,22841,47,134461,15686,106917,179494,355173,59,1220075,63,
%U 1593189,1952705,2228909,631005,13778268,75,9962313,20732901,34805473,83,113693883
%N a(n) = Sum_{d|n} d * sigma(n/d, d).
%C Here sigma(n,k) is the sum of the k-th powers of the divisors of n.
%C Logarithmic derivative of A198301.
%F L.g.f.: Sum_{n>=1} Sum_{k>=1} sigma(n,k) * x^(n*k)/n.
%e 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 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 + ...
%t a[n_] := DivisorSum[n, # * DivisorSigma[#, n/#] &]; Array[a, 40] (* _Amiram Eldar_, Aug 18 2023 *)
%o (PARI) {a(n)=sumdiv(n, d, d*sigma(n/d,d))}
%o (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 Cf. A198301 (exp), A185301, A198299.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 27 2012
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