2 Edited to point out the mistake I made.

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.

1

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.