I have been studying this sequence ([A266882](https://oeis.org/A266882) in the OEIS) and found the following pattern:


$13 + 17 + 19 + 23 + 29 = 101$ (101 is prime)


37 does not hold.


$223 + 227 + 229 + 233 + 239 = 1151$ (1151 is prime)


The same is true for 1087, 1423, 1483, and 2683.


So, is 37 the only prime of this sequence (A266882) for whom the sum is not also a prime?



Primes p(n) such that p(n) + p(n+3) = p(n+1) + p(n+2) and p(n) + p(n+4) = p(n+2) + p(n+3) - Conjecture
No. A simple check shows that it is already false for the next few primes in the sequence, namely for 4783, 6079, 7331.
In fact, one could even ask whether these first numbers are the only ones for which this sum is prime, but it turns out that it later holds also for 11057 and 12269. 


