Proves density-one lower bound g_k(n) ≥ (3(k-1)/log 12 - ε) log n for almost all n and pointwise upper bound g_k(n) ≤ (k-1)log2 n + log2 log n + O_k(1).
Pomerance,Divisors of the middle binomial coefficient, Amer
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Prime-Power Rarefaction and a Density-One Lower Bound for Erd\H{o}s Problem 400
Proves density-one lower bound g_k(n) ≥ (3(k-1)/log 12 - ε) log n for almost all n and pointwise upper bound g_k(n) ≤ (k-1)log2 n + log2 log n + O_k(1).