question in prime numbers Is it true that in any successive (natural) $2p_n$ numbers there are at least three numbers  that are not divisible by any prime less (not equal) than $p_n$? Here, $p_n$ denotes the $n$-th prime number. 
For
example in any six successive numbers there are at least 3 numbers that are not divisible by 2,in any 10 successive numbers there are 3 numbers that are not divisible by 2 or 3, in any 14 successive numbers there are at least 3 that are not divisible by 2, 3, or 5.
 A: I believe the answer is no.  In 1931 Erik Westzynthius showed that there are arbitrarily large gaps in the sequence S(n) of numbers which are relatively prime to the the first n primes.  He not only showed that there were gaps of size 2*(p_{n-1}), which would give rise to your conjecture, he also showed that for large n ( greater than something like e^(e^e) ) that the gaps got larger than p_n*f(n) where f(n) is an increasing function involving log(n) and log(log(log(n))), which says that there are long runs of numbers each of which has one of the primes below and including p_n as a factor, and thus gives a negative answer to your question.
It is not known for what n this first occurs (a sequence of length 2*p_n consecutive numbers all of which have a factor among the first n primes).  I am working on an upper bound for f(n) currently.  For more information, see the related question Erik Westzynthius's cool upper bound argument: update? .  I hope to post some information there soon.
Gerhard "Ask Me About System Design" Paseman, 2010.12.14
A: EDIT: The following is false. The problem is that we only get an asymptotic for the number of such integers in the interval $[0,N]$, and to examine an interval of size $2p_n$ we need to examine the difference between this and $[0,N+2p_n]$. I was using the lower bound for this latter quantity as an upper bound, hence the erroneous result below. In reality, the answer could well be zero infinitely often, as Gerhard Paseman indicates.
For sufficiently large intervals, this is true; it follows from Buchstab's theorem (see Montgomery and Vaughan chapter 7.2) that, for sufficiently large $N$, there are approximately
$$ e^{-\gamma}\frac{2p_n}{\log p_n} $$
numbers in the interval $[N,N+2p_n]$ with all prime divisors at least $p_n$. For all primes at least 7, this is larger than 3. The cases 2,3 and 5 can be checked directly, as you mention in your question.
I imagine that if you were interested, it should be fairly easy to work out a specific (albeit large) lower bound for something like the above to hold, and then run a computer search on all smaller intervals.
Of course, there may well be a simple elementary way anyway, which I can't see right now.
