Primes that satisfy p(k+1) - p(k) > p(k+3) - p(k+1), where p(k) is the k-th prime.

89, 113, 139, 181, 211, 293, 337, 421, 449, 523, 631, 661, 811, 839, 863, 887, 953, 997, 1021, 1051, 1069, 1129, 1201, 1259, 1307, 1327, 1409, 1439, 1471, 1531, 1583, 1637, 1669, 1759, 1847, 1951, 2069, 2113, 2179, 2221, 2251, 2311, 2357, 2423, 2557, 2593

1

Pintz proves (as part of a much more general theorem) that an infinite number of primes are in this sequence. It was conjectured by Erdos, Polya, and Turan 66 years ago.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Janos Pintz, On a conjecture of Erdos, Polya and Turan on consecutive gaps between primes, arXiv 1504.06860 (Apr 26 2015)

(Mma) Prime[Select[Range[1000], Prime[# + 1] - Prime[#] > Prime[# + 3] - Prime[# + 1] &]]

nonn

T. D. Noe, Apr 27 2015