MathNet 2016-part-01 — 184 problems (EN–VI) Question 147

Determine all positive integers $M$ for which the sequence $a_0, a_1, a_2, \ldots$, defined by $a_0 = \frac{2M+1}{2}$ and $a_{k+1} = a_k[a_k]$ for $k = 0, 1, 2, \ldots$, contains at least one integer term.