An arithmetic sequence of numbers without certain prime factors I just attended a lecture at my school about prime numbers and the idea of being coprime and what kind of applications that has, and this question popped up in my head and I thought about it for a while and asked a few friends and wasn't quite sure what to think. One of my friends told me about this website so I thought I would give it a shot. So here's the question:
What is the largest arithmetic sequence (most amount of terms) such that none of its terms are divisible by the first $n$ prime numbers, and which all of its terms are less than $n\#$ (where $\#$ denotes the primorial function)?
For example, for $n = 2$ the largest such sequence would be $1, 5$ and for $n = 3$ the largest such sequence would be $1, 7, 13, 19$ or $11, 17, 23, 29$.

In addition to what Patrick said, the only available terms that are the products of two primes $p, q$ (or $1$) where $p, q > p_n$. So for $4\#$ you would have $1, 11, 11 \times 11, 11 \times 13, 11 \times 17, 11 \times 19, 13, 13 \times 13, 17, 19, 23, 27,$ (along with the rest of primes less than $210$).
In its current state, it looks like these arithmetic progressions are going to be awfully low in number, and depend on arithmetic progressions found in prime numbers, which is pretty low as well. More about primes in arithmetic progression: http://en.wikipedia.org/wiki/Primes_in_arithmetic_progression http://mathworld.wolfram.com/PrimeArithmeticProgression.html
Not a full answer by any means, but some guiding tips.
