When $A_n = n$, the function $f$ defined above counts the number of odd divisors of $n$.

Let $A=\{a_1 < \cdots\}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to $$n=\sum_{u\leq i\leq v}a_i.$$ Is there an $A$ such that $f(n)\geq 2$ for all large $n$?

This also follows from Tao's construction with $f(n) \gg \log n$ [Ta26].