Upper Bound on Aliquot Sequence

Question

Let $s_1(n)=\big{(}\sum_{d|n}d\big{)}-n=\sigma_1(n)-n$ be the restricted divisor sum, and define $s_k(n)=s_1(s_{k-1}(n))$ as the $k^{th}$ term of the aliquot sequence starting at $n$. What is the best proven upper bound on $s_k(n)$? In other words, if the sequence starting at $n$ seems to tend to infinity, how fast does it do so? Existing bounds on $\sigma(n)$ might help... Alternatively, is it possible to bound $\sum_{n\leq x}s_k(n)$?

Answer 1

Erdos, On asymptotic properties of aliquot sequences, Math Comp 30 (1976), no. 135, 641-645, MR0404115 (53 #7919), proved that for every fixed $k$ and every $\delta>0$ and for all $n$ except a sequence of density zero one has $$(1-\delta)n(s(n)/n)^i<s_i(n)<(1+\delta)n(s(n)/n)^i$$ for $1\le i\le k$.