Least prime p such that the interval p to p + floor(log(p)^2) contains at least n primes.
2, 5, 13, 37, 59, 89, 113, 173, 431, 431, 1259, 1277, 1277, 2657, 2657, 13669, 19373, 19373, 27733, 106619, 113039, 113039, 115751, 524857, 560081, 783619, 1219717, 1652731, 2704901, 2704901, 8756939, 19800101, 21401987, 40268381, 40268387, 40268441, 167540111, 286352189, 311773001, 311773001, 311773001
1
No other primes less than 10^9. Similar to S000548.
T. D. Noe, Plot of 41 terms
K. Soundararajan, The distribution of prime numbers, arXiv math 0606408 (Jun 16 2006).
(Mma) mx = 0; t = {}; Do[p = Prime[n]; q = PrimePi[p + Floor[2*Exp[-EulerGamma]*Log[p]^2]] - n + 1; If[q > mx, Do[AppendTo[t, p], {q - mx}]; mx = q], {n, 20000}]; t
Cf. A120934 (similar, but with power 1), S000547, S000548, S000578.
nonn,hard,nice
T. D. Noe, Apr 15 2015