# Which progressions and sequences are guaranteed to contain infinitely many primes?

Progressions. Dirichlet famously proved that, if $a$ and $b$ are coprime integers, the arithmetic progression $a+db$ (i.e., with $d=1,2,3,\dots$) contains infinitely many prime numbers. Are there any other such results? I believe there are no analogous quadratic or higher-power sequences, but I could have missed some.

Sequences. Other than the obvious/trivial ones (or arithmetic progressions, covered above), which number sequences [with a regular algebraic definition] contain infinitely many prime numbers?

• You can construct infinitely many such sequences. OEIS contains some of the interesting ones. – daniel Oct 19 '15 at 0:03
• @daniel: Do you have any specific references? I've only found trivial ones (e.g., Sophie Germain primes <oeis.org/A005384>), or ones using functions like floor (e.g., oeis.org/A132222). I'm hoping for sequences/progressions with simpler definitions, like Dirichlet's. – Kieren MacMillan Oct 19 '15 at 1:23