Question 26
Let $n \in \mathbb{N}$ and $t_{1}, t_{2}, \ldots, t_{k}$ be different positive divisors of $n$. An identity of the form $n = t_{1} + t_{2} + \ldots + t_{k}$ is called a representation of $n$ as a sum of different divisors. Two such representations are considered equal if they differ only in the order of the summands (for example, $20 = 10 + 5 + 4 + 1$ and $20 = 5 + 1 + 10 + 4$ are twice the same representation of $20$ as the sum of different divisors). Let $a(n)$ be the number of different representations of $n$ as a sum of different divisors. Show or disprove:
There is a $M \in \mathbb{N}$ with $a(n) \leq M$ for all $n \in \mathbb{N}$.
There is a $M \in \mathbb{N}$ with $a(n) \leq M$ for all $n \in \mathbb{N}$.