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Iterated aliquot sums
Related to case (b) above is a system which repeats the replacement n Apply[Plus, Divisors[n]] - n or equivalently n DivisorSigma[1, n] - n .

Properties [of number theoretic sequences]
(a) The number of divisors of n is given by DivisorSigma[0, n] , equal to Length[Divisors[n]] . … (b) (Aliquot sums) The quantity that is plotted is DivisorSigma[1, n] - 2n , equal to Apply[Plus, Divisors[n]] - 2n . … For large n , DivisorSigma[1, n] is known to grow at most like Log[Log[n]] n Exp[EulerGamma] , and on average like π 2 /6 n (see page 1093 ).

Perfect numbers
Perfect numbers with the property that Apply[Plus, Divisors[n]] 2n have been studied since at least the time of Pythagoras around 500 BC. … Various generalizations of perfect numbers have been considered, requiring for example IntegerQ[DivisorSigma[1, n]/n] (pluperfect) or Abs[DivisorSigma[1, n] - 2n] < r (quasiperfect).