This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A120934 #48 Feb 16 2025 08:33:02
%S A120934 2,11,457,3251,165701,10526557,495233351,196039655873,10687033762033,
%T A120934 79006533276941,4313367040646743,1740318019946551931
%N A120934 Least prime p such that the interval [p,p+log(p)] contains n primes.
%C A120934 Soundararajan states that, on average, there is one prime in the interval [k,k+log(k)] for any number k. Is there an upper limit to the number of primes in such an interval? Not if the prime k-tuple conjecture is true, in which case a(n) exists for all n. Note that a(n) > e^A008407(n). See A120935 for the largest prime in the interval.
%C A120934 a(n) begins a sequence of n primes whose prime pattern is one of the patterns in the n-th row of A186634. For example, the sequence of four consecutive primes beginning with 3251 is (3251, 3253, 3257, 3259), which has pattern (0, 2, 6, 8), which is in the 4th row of A186634.
%H A120934 K. Soundararajan, <a href="https://arxiv.org/abs/math/0606408">The distribution of prime numbers</a>, arXiv:math/0606408 [math.NT], 2006.
%H A120934 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-tupleConjecture.html">k-Tuple Conjecture</a>
%F A120934 This sequence grows superexponentially; a weak lower bound is a(n) >> (log n)^n. It seems that a(n) > n^n. - _Charles R Greathouse IV_, Apr 18 2012
%F A120934 A lower bound is a(n) > e^A008407(n). a(n) < b*e^A008407(n), for 2 <= n <= 12, b < 1.49. For 9 <= n <= 12, b < 1.0006. a(13) > 701673591209763173865. - _Florian Baur_, Jul 12 2023 [Corrected by _Pontus von Brömssen_, Nov 12 2023]
%e A120934 a(2)=11 because p=11 is the first prime with log(p) > 2 and 11+2 is prime.
%t A120934 i=1; Table[While[p=Prime[i]; PrimePi[p+Log[p]]-PrimePi[p]+1< n, i++ ]; p, {n,5}]
%o A120934 (PARI)
%o A120934 A120934(n) = {
%o A120934 my(v = vector(n,k,prime(k)), i = 1);
%o A120934 while(v[(i - 2) % n + 1] - v[i] > floor(log(v[i])),
%o A120934 v[i] = nextprime(v[(i - 2) % n + 1] + 1);
%o A120934 i = i % n + 1; );
%o A120934 return(vecsort(v)[1]) } \\ _Florian Baur_, Jul 12 2023
%Y A120934 Cf. A120935, A120936 (number of primes in the interval [n, n+log(n)]), A020497.
%Y A120934 Cf. A008407, A186634.
%K A120934 hard,more,nonn,nice
%O A120934 1,1
%A A120934 _T. D. Noe_, Jul 21 2006
%E A120934 a(12) from _Donovan Johnson_, Apr 18 2012