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A320940 a(n) = Sum_{d|n} d*sigma_n(d).

%I A320940 #26 Sep 08 2022 08:46:23
%S A320940 1,11,85,1127,15631,287021,5764809,135007759,3487020610,100146496681,
%T A320940 3138428376733,107032667155169,3937376385699303,155582338242604221,
%U A320940 6568408966322733475,295154660699054931999,14063084452067724991027,708239400347943609329270
%N A320940 a(n) = Sum_{d|n} d*sigma_n(d).
%H A320940 Seiichi Manyama, <a href="/A320940/b320940.txt">Table of n, a(n) for n = 1..385</a>
%H A320940 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A320940 a(n) = n * [x^n] -log(Product_{k>=1} (1 - x^k)^sigma_n(k)).
%F A320940 a(n) = Sum_{d|n} d^(n+1)*sigma_1(n/d).
%F A320940 a(n) ~ n^(n+1). - _Vaclav Kotesovec_, Feb 16 2020
%e A320940 a(6) = 1*sigma_6(1)+2*sigma_6(2)+3*sigma_6(3)+6*sigma_6(6) = 1+2*65+3*730+6*47450 = 287021.
%p A320940 with(numtheory): seq(coeff(series(n*(-log(mul((1-x^k)^sigma[n](k),k=1..n))),x,n+1), x, n), n = 1 .. 20); # _Muniru A Asiru_, Oct 28 2018
%t A320940 Table[Sum[d DivisorSigma[n, d], {d, Divisors[n]}] , {n, 18}]
%t A320940 Table[n SeriesCoefficient[-Log[Product[(1 - x^k)^DivisorSigma[n, k], {k, 1, n}]], {x, 0, n}], {n, 18}]
%o A320940 (PARI) a(n) = sumdiv(n, d, d*sigma(d, n)); \\ _Michel Marcus_, Oct 28 2018
%o A320940 (Python)
%o A320940 from sympy import divisor_sigma, divisors
%o A320940 def A320940(n):
%o A320940     return sum(divisor_sigma(d)*(n//d)**(n+1) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Feb 15 2020
%o A320940 (Magma) [&+[d*DivisorSigma(n,d):d in Divisors(n)]:n in [1..18]]; // _Marius A. Burtea_, Feb 15 2020
%Y A320940 Cf. A000203, A001001, A027847, A027848, A319647, A321141.
%K A320940 nonn
%O A320940 1,2
%A A320940 _Ilya Gutkovskiy_, Oct 28 2018