Let $A=\{a_1<\cdots \}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to$$n=\sum _{u\le i\le v}{a}_i.$$Is there such an $A$ for which $f(n)\to \infty$ as $n\to \infty$? Or even where $f(n)\ge 2$ for all large $n$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write the partial sums [ s_0:=0,\qquad s_k:=a_1+\cdots+a_k\ \ (k\ge 1). ] Then [ \sum_{u\le i\le v} a_i = s_v-s_{u-1}, ] so $f(n)$ is exactly the number of pairs $(i,j)$ with (0\le i<j) such that [ n=s_j-s_i. ] Equivalently, if (S={s_0,s_1,s_2,\dots}), then $f(n)$ is the multiplicity of $n$ in the difference multiset $…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.