Estimate $n_{k}$ , the smallest integer $> 2 k$ such that $\prod_{1 \leq i \leq k} (n_{k} - i)$ has no prime factor in $(k, 2 k)$ .

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number theory · open · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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

This is Erdős’s Problem #451 and (as far as is currently recorded) it remains open in the sense that no rigorous asymptotic for (n_k) is known. ([Erdős Problems][1])

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

oeis

A386620 — a(n) is the smallest integer k > 2*n such that Product_{i=1..n} (k - i) has no prime factor p in n < p < 2*n.3,6,9,20,13,21,21,22,65,220,51,338,133,321,339,340,113,114,368,550,805,2691,1884,2664,7653,7654,36887,36888,21234,21235,

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