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Much progress has been made in six months. See A225423.
T. D. Noe, Table of n, a(n) for n = 1..1000
James Maynard, Small gaps between primes, arxiv 1311.4600, Nov 19 2013.
Select[Prime[Range], PrimeQ[600 + #] &]
Cf. A001359 (first member of a twin prime pair), A028334, A225423.
Sequence in context: A079699 A053176 A032669 * A147603 A106084 A110053
Adjacent sequences: A231604 A231605 A231606 * A231608 A231609 A231610
T. D. Noe, Nov 21 2013