Claims

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1976_01_01_erdos: Display (6) of Erdős's 1976 paper in Publ. Math. Debrecen, proved by a smooth-number count, gives log k(n) >= (1/sqrt 2 - o(1)) sqrt(log n log log n); a refereed lower bound, while both questions of the problem stay open.

1976_01_01_erdos_upper: Erdős's 1976 statement, without proof, that he can show n_k > k^2 exp((log k)^c) for some c > 0, which gives k(n) <= n^{1/2} exp(-(log n)^{c'}); claimed, since no proof is printed or referenced.

2025_12_28_tang: A two-page note by Quanyu Tang, posted to the site's thread on 28 December 2025, proving log k(n) >= (1/sqrt 2 - o(1)) sqrt(log n log log n) by a pigeonhole count of smooth integers; not refereed.

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