Decomposing a Factorial into Large Factors

I’ve just uploaded to the arXiv the paper “Decomposing a factorial into large factors“. This paper studies the quantity , defined as the largest quantity such that it is possible to factorize into factors , each of which is at least . The first few values of this sequence are OEIS A034258 ). For instance, we have, because on the one hand we can factor but on the other hand it is not possible to factorizeinto nine factors, each of which isor higher. This quantity was introduced by Erdös, who asked for upper and lower bounds on ; informally, this asks how equitably one can split up into factors. When factoring an arbitrary number, this is essentially a variant of the notorious knapsack problem (after taking logarithms), but one can hope that the specific structure of the factorial can make this particular knapsack-type problem more tractable. Since for any putative factorization, we obtain an upper bound thanks to the Stirling approximation. At one point, Erdös, Selfridge, and Straus ... Read full article.