Prime Blocks in A007310: The Unique Run of Length 7

We study maximalblocks of consecutive primeterms in A007310, the sequence of positive integers congruent to 1 or 5 modulo 6. We prove that the initial block 5,7,11,13,17,19,23 is the unique prime block of length 7, and that no longer prime block exists. For each r≥1, we define Pr as the sequence of r-th terms of the prime blocks. As consequences, P7 = {23} and Pr = ∅ for all r≥8.