S000548


Least prime p such that the interval p to p + floor(log(p)^2) contains exactly n primes.

2, 5, 13, 37, 59, 89, 173, 433, 431, 1259, 1279, 1277, 2659, 2657, 13669, 19391, 19373, 27733, 106619, 113041, 113039, 115751, 524857, 560081, 783619, 1219717, 1652731, 2704903, 2704901, 8756939, 19800101, 21401987, 40268381, 40268387, 40268447, 40268441

1

S000548

Related to Cramer’s conjecture. This is the least p that reaches height n in the graph of S000547. The last p of height n > 2 appears to be undefined; the last height of 1 appears to be 7 and the last height of 2 appears to be 19. See S000577 for a slightly different version.

T. D. Noe, Plot of 47 terms

T. D. Noe, Table of 47 terms

Harald Cramer, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2, 23-46, 1936.

Eric W. Weisstein, MathWorld: Cramér Conjecture

(Mma) nn = 20; t = Table[0, {nn}]; p = 1; n = 0; toDo = nn; While[toDo > 0, n++; p = NextPrime[p]; prms = PrimePi[p + Floor[Log[p]^2]] - n + 1; If[prms <= nn && t[[prms]] == 0, t[[prms]] = p; toDo--]]; t

Cf. S000547, S000577.

nonn,hard

T. D. Noe, Mar 24 2015

© Tony D Noe 2014-2015