Let $h(n)$ count the number of powerful (if $p\mid m$ then $p^2\mid m$) integers in $[n^2,(n+1{)}^2)$. Estimate $h(n)$. In particular is there some constant $c>0$ such that$$h(n)<(\log n{)}^{c+o(1)}$$and, for infinitely many $n$,$$h(n)>(\log n{)}^{c-o(1)}?$$

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

Write (F_2) for the squarefull (powerful) integers and (Q(x):=|\\{m\le x:m\in F_2\\}|). A standard fact is that every squarefull integer has a unique representation [ m=a^2b^3\qquad(b\ \text{squarefree}), ] and Erdős–Szekeres proved (with an explicit constant) [ Q(x)=c_2 x^{1/2}+O(x^{1/3}),\qquad c_2=\frac{\zeta(3/2)}{…

llm-hunter · gpt pro 5.2 · unverified

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