From: Dave Rusin <rusin@math.niu.edu>
Subject: Re: Slightly Composites.-Asymptotic formula
Date: Mon, 22 Oct 2001 23:03:53 -0500 (CDT)
Newsgroups: sci.math.research
To: Lord_Bern@hotmail.com
Summary: Distribution of numbers with two prime factors

In article <8f423bnrb2kb@legacy> you write:
>Let an integer be slightly composites if it is of the form pq where p
>and q are primes. A well known result is that 
>#Slightly composites less than n ~ (n * loglogn)/log n. Does anyone
>know a good lower bound for the the number of slightly composites less
>than n? 

Not sure exactly how good you want this. Clearly (taking  p <= q) the
number you want is the sum over all primes  p  not exceeding  sqrt(n)  of
pi(n/p), which (summand) can be bounded below by something of the form
(1 - eps) (n/p) / log(n/p)  since  n/p  will be large (at least  sqrt(n). )
Note that the denominator is  log(n) - log(p)  which is between (1/2) and (1)
times  log(n)  itself, so although I could probably tighten this up, we
at least can bound the summand by something like  (1/2)(1/p)(n/log(n)).
Then you need only the sum of  1/p  as  p  ranges over the primes less than 
sqrt(n), which is known to be something like  log(log(sqrt(n))) = log(log(n))-
log(2). So overall I can get within about a factor of  2  of what you
consider to be the estimated value of your count. Is that good enough?

dave

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