If N has more divisors than any lesser number, N is "highly composite." http://oeis.org/A002182 I conjecture that the number d(N) of divisors of any highly composite N always obeys d(N) >= 2^Li(ln(N)). Results of S.Ramanuan: Highly composite numbers, Proc. London Math. Soc. (2) 14 (1915) 347-409. http://ramanujan.sirinudi.org/Volumes/published/ram15.pdf show that my conjecture is true at least for an infinite subset of the highly composite N. It looks plausible that by similar techniques you can prove it for all sufficiently large highly composite N, although as far as I know nobody has done so. If so, and if we knew exactly what "sufficiently large" meant, that'd reduce the conjecture to a finite computation. Which then might be within reach of computer (e.g. at least the first 120000 highly composite numbers are known). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)