The indices of the primes in S000190.
1, 2, 4, 8, 15, 21, 30, 46, 61, 91, 121, 154, 189, 217, 258, 409, 548, 590, 775, 832, 906, 1227, 1270, 1570, 1847, 2116, 2688, 3607, 3714, 3990, 4483, 4502, 6041, 6045, 8078, 9255, 9833, 12136, 12226, 13208, 14356, 16964, 17511, 18858, 18974, 20476, 23489, 23846
1
It appears that s(n+1) <= 2*s(n).
T. D. Noe, Plot of 200 terms
T. D. Noe, Table of 200 terms
Edward Tutaj, Prime numbers with a certain extremal type property, arXiv 1408.3609, Aug 15 2014
(Mma) t = {2}; Do[k = PrimePi[t[[-1]]]; s = Table[(n - k)/(Prime[n] - Prime[k]), {n, k + 1, 3*k}]; pos = Position[s, Max[s]][[-1]]; AppendTo[t, Prime[k + pos[[1]]]], {99}]; PrimePi[t]
nonn
T. D. Noe, Aug 19 2014