Take any number N greater than 2P, will there be always be a number that isn't a multiple of 2,3...P between N and N+2P

Question

I was working on the Bertrand's postulate(which states that there's always a prime between any integer n and 2n), and I wonder if it won't only work for 1,2,3...2P for set of primes (2,3,5...P), but also for integers N, N+1, N+2....N+2P for set of primes (2,3,5...P). I checked so far for numbers bigger than 2P, say N. And I checked for numbers between N and N+2P, and I always found a number relatvely prime to 2,3,...P. For example say, (2,3,5,7), let N be 20, between 20 and 34, there are 3 numbers, 23, 29, and 31 that are relatively primes to (2,3,5,7). Now will this be true for numbers between N and N+2P using primes (2,3,5...P)?