The iterated sum of divisors function $\sigma^k(n)$ is $a_k$ in the recurrence relation $a_0 = n$ and $a_i = \sigma(a_{i - 1})$ for $i > 0$ where $\sigma(x)$ is the sum of divisors function.

Since $n$ itself is included in the set of its divisors, the sequence generated by repeated iterations is an increasing sequence (that is, in ascending order). For example, iterating the sum of divisors function for $n = 2$ gives the sequence 2, 3, 4, 7, 8, 15, etc. Erdos conjectured that there is a limit for $(\sigma^k(n))^{\frac{1}{k}}$ as $k$ approaches infinity.

Bibliography

1

R. K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: B9