From: blackj@seal.cs.ucdavis.edu (John Black)
Subject: Re: Easy (hopefully) Number Theory  Question
Date: 9 Mar 1999 08:46:40 GMT
Newsgroups: sci.math
Keywords: Highly composite numbers and bounds on number of divisors

Gerry Myerson (gerry@mpce.mq.edu.au) wrote:
: In article <7bq5vc$qbv$1@mark.ucdavis.edu>, blackj@seal.cs.ucdavis.edu
: (John Black) wrote:

: > I want to find numbers with lots of divisors.  I fix a range, R = [1,m] and
: > ask, which x \in R has the largest number of divisors.  I am interested
: > in a general formula (or a bound) on the number of divisors of such an x.

: Theorem 317 of Hardy & Wright, The Theory of Numbers, says that if e > 0 
: then d(n) < 2^{(1 + e) log n / log log n} for all n > n_0 (e), 
: and d(n) > 2^{(1 - e) log n / log log n} for infinitely many n. 
: Here, d(n) is the number of divisors of n. 

Thanks to everyone who replied.  I found several bounds like the above; 
the principal investigators were apparently Ramanujan, who termed these
things "Highly Composite Numbers" in a 1915 paper, and Nicholas, who improved
the bounds to the current best-known (as far as I can tell).  I won't write
the Nicholas bound because it's a big mess.

I worked out for myself the integers which are minimum and have 2^k divisors
for k = 1, 2, ....  This was a rather cute problem, but I today found out
that Ramanujan had already worked all this out in his 1915 paper.

Thanks again to all!

john//