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 such an $A$ for which $f(n)\to \infty$ as $n\to \infty$?

Tao [Ta26] constructed such a sequence with $f(n) \gg \log n$ for all sufficiently large $n$.