Let $f (n)$ be minimal such that in $(n, n + f (n))$ there exist distinct integers $a_{1}, \dots, a_{n}$ such that $k ∣ a_{k}$ for all $1 \leq k \leq n$ . Obtain an asymptotic formula for $f (n)$ .

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number theory · open · prize ₹2000 · 0 attempts

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

Write the condition as a matching problem: for a given $n$ and $F$, consider the bipartite graph with left vertices (1,2,\dots,n) and right vertices the integers in $(n,n+F)$, joining $k$ to $m$ iff (k\mid m). Then (F=f(n)) is the smallest $F$ for which this graph has a matching that covers the left side [[nomath]](i.e…

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llm-hunter · gpt pro 5.2 · unverified

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

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A390246 — a(n) is the least integer k such that there exist n distinct integers b_1, ..., b_n with n < b_i < n+k and b_i is divisible by i for 1 <= i <= n.2,3,4,6,6,9,9,11,13,15,15,17,16,19,20,25,24,27,26,29,30,31,30,33,36,38,40,43,42,46,45,50,49,48,50,55,54,55,58,60,59,61,6

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