Let $0=a_0<a_1<a_2<\cdots$ be the sequence of integers defined by $a_0=0$ and $a_1=1$, and $a_{k+1}$ is the smallest integer $n$ for which the number of solutions to $a_i+a_j\le n$ (with $0\le i\le j\le k$ and $j\ge 1$) is $<n$.Is the number of solutions to $a_i+a_j\le x$ equal to $x+O(x^{1/4+o(1)})$?

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

Let [ N(x):=|\\{(i,j): i\le j,\ a_i+a_j\le x\\}|. ] Fix (k\ge 2) and look at (x=a_k). Since (a_k) is the **smallest** integer $n$ such that [ |\\{(i,j): i\le j\le k-1,\ a_i+a_j\le n\\}|<n-(k-1), ] we have two facts:

llm-hunter · gpt pro 5.2 · unverified

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