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 A381044 #33 May 27 2025 10:10:04
%S A381044 41,59,71,101,107,137,149,179,191,197,227,239,269,281,311,347,419,431,
%T A381044 461,521,569,599,617,641,659,809,821,827,857,881,1019,1031,1049,1061,
%U A381044 1091,1151,1229,1277,1279,1289,1297,1301,1303,1319,1423,1427,1429,1447,1451,1481,1483,1487,1489
%N A381044 Primes prime(k) followed by a gap, prime(k+1)-prime(k), smaller than the local geometric average gap between consecutive primes: log(prime(k))/e^(gamma).
%C A381044 Primes prime(k) such that log(prime(k+1)-prime(k)) < log(log(prime(k)))-gamma, where log is the natural logarithm and gamma is Euler’s constant (A001620).
%C A381044 Except for terms less than 41, A001359 (Lesser of twin primes) is a subsequence. From 41, the first term not included is 1279.
%C A381044 It has been conjectured that primes are distributed around their average spacing in a Poisson distribution (cf. D. A. Goldston and A. H. Ledoan). This is the basis of the conjecture that, for k tending to infinity, the asymptotic limit of the average of log(prime(k+1)-prime(k)) is log(log(prime(k))) - gamma (where gamma is Euler's constant). Also, the geometric mean of the gap between consecutive primes [p(k+1)-p(k)] is equivalent to log(prime(k)) / e^gamma.
%H A381044 D. A. Goldston and A. H. Ledoan, <a href="https://arxiv.org/abs/1111.3380">On the differences between consecutive prime numbers, I</a>, arXiv:1111.3380 [math.NT], 2011-2012.
%H A381044 Carlos Rivera, <a href="https://www.primepuzzles.net/conjectures/conj_082.htm">Conjecture 82. Average of log Dn / log(logPn) equal R = 0,877 08...</a>, The Prime Puzzles & Problems Connection.
%F A381044 Limit_{n->oo} n / PrimePi(a(n)) = 1-e^(-1/e^gamma).
%e A381044 29 is not a term because log(31-29) > log(log(29))-0.5772156649, i.e.: 0.693147 > 0.636894.
%e A381044 41 is a term because log(43-41) < log(log(41))-0.5772156649, i.e.: 0.693147 < 0.734779.
%t A381044 Select[Prime[Range[237]],Log[NextPrime[#]-#]<Log[Log[#]]-EulerGamma&] (* _James C. McMahon_, May 02 2025 *)
%o A381044 (PARI) forprime(P=3, 1500, my(Q=nextprime(P+1), LNDP=log(Q-P)); if(LNDP<log(log(P))-Euler, print1(P,", ")));
%Y A381044 Cf. A001359, A046869, A001223, A001620.
%K A381044 nonn
%O A381044 1,1
%A A381044 _Alain Rocchelli_, Apr 14 2025