Primes p at which (nextprime(p) - p) / (p - prevprime(p)) is a new record high or a record low.
3, 5, 11, 29, 31, 113, 127, 139, 149, 199, 521, 523, 1151, 1669, 2971, 2999, 6947, 7759, 10007, 12163, 16141, 25301, 25471, 28277, 35729, 40639, 79699, 102761, 149629, 173359, 173429, 265621, 371027, 404851, 461801, 576881, 838249, 838349, 1349533, 1562051
1
In the paper below, Pintz proves theorems about these numbers, but does not display them. For the primes shown, with the exception of 113 and 127, they are part of a twin-prime pair.
T. D. Noe, Plot of 62 terms
T. D. Noe, Table of 62 terms
Janos Pintz, On the ratio of consecutive gaps between primes, arXiv 1406.2658, Jun 10 2014.
(Mma) mn = 10; mx = 0; tMn = {}; tMx = {}; Do[f = (Prime[n + 1] - Prime[n])/(Prime[n] - Prime[n - 1]); If[f < mn, mn = f; AppendTo[tMn, {Prime[n], f}]]; If[f > mx, mx = f; AppendTo[tMx, {Prime[n], f}]], {n, 2, 1000000}]; Union[Transpose[tMx][[1]], Transpose[tMn][[1]]]
Cf. S000080 (record highs), S000081 (record lows).
nonn
T. D. Noe, Jun 12 2014