This is related to an older question about prime k-tuples and constellations, but takes a slightly different direction.

Given an integer k, we want to find n such that the interval {n+1, ..., n+k} contains as many primes as possible. (We consider only n ≥ k to eliminate certain exceptional cases, such as {3,5,7}, which is irregular since for k=5 there can be at most 2 primes in the interval if n>2.)

There is an obvious upper bound a_{k} on the number of primes in this interval, given by considering the numbers modulo p for all primes p ≤ k. More precisely, a_{k} is the largest possible cardinality of a set A ⊂ {1, ..., k} such that for some n, the set n+A does not contain any numbers divisible by any prime p ≤ k.

For example, a_{3}=2 since out of 3 consecutive numbers at least one is even, and a_{7}=3 since out of seven consecutive numbers at least three are even, and at least one of the odd numbers is divisible by 3. Similarly, a_{9}=4 and a_{13}=5.

The four bounds listed so far can be achieved by taking n=4, n=10, n=10, n=36, respectively. The natural question to ask is whether for every k there is a value of n such that the interval {n+1, ..., n+k} contains a_{k} primes, but the comments on the older question make me suspect that this may be open. (Another natural question is whether there are infinitely many such n, but since for k=3 this is the twin prime conjecture, that's definitely out of reach at present.)

Since the natural question to ask seems very hard, my question instead is this: Is anything about the asymptotics of this problem? More precisely, I'd like to know if something like the following statement is true: "For every ε>0, there exist infinitely many pairs (k,n) such that the interval {n+1, ..., n+k} contains at least (1-ε)a_{k} prime numbers."

There are a few different ways to tweak that statement -- for example, we could ask for infinitely many n for a fixed k, or we could let both k and n become arbitrarily large. (Of course k will need to become arbitrarily large as ε becomes small.) I'd be happy with any of them -- I'm asking this question out of curiosity rather than out of a need for a specific result.