Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #9 Sep 10 2019 19:55:58
%S 0,1,1,5,1,18,1,33,28,58,1,246,1,178,369,577,1,1539,1,2774,2531,2170,
%T 1,16706,3126,8362,20413,35366,1,116444,1,135425,178479,131362,94933,
%U 1110999,1,524650,1596521,2530946,1,7280892,1,8403734,16364457,8389138,1,78568322,823544,43420683
%N Expansion of Sum_{k>=1} k^2 * x^(2*k) / (1 - k * x^k).
%F a(n) = Sum_{d|n, d>1} (n/d)^d = Sum_{d|n, d<n} d^(n/d).
%F a(p) = 1, where p is prime.
%F a(n) = A055225(n) - n.
%t nmax = 50; CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t Table[DivisorSum[n, (n/#)^# &, # > 1 &], {n, 1, 50}]
%o (PARI) a(n)={sumdiv(n, d, if(d > 1, (n/d)^d))} \\ _Andrew Howroyd_, Sep 10 2019
%Y Cf. A055225, A078308, A087909.
%K nonn
%O 1,4
%A _Ilya Gutkovskiy_, Sep 10 2019