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 A274123 #36 Apr 02 2024 14:34:03
%S A274123 2,127,149,383,431,443,487,557
%N A274123 Let F(g,p) be the frequency of g up to the prime nextprime(p+1). F(g,p_i) is a record for some prime p_i and F(g, p_(i+1)) is a new record for the next larger prime after p_i. The sequence lists the primes p_(i+1), except a(1) = 2.
%C A274123 Up to large values of n, 6 is conjectured to be the most occurring gap. See link "Polignac's conjecture". If this conjecture is true the sequence is finite.
%C A274123 For primes up to 10^8, there are no more terms. Up to 10^6, the prime gap 2 occurs 8169 times, the gap 4 occurs 8143 times and the gap 6 occurs 13549 times.
%H A274123 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polignac%27s_conjecture">Polignac's conjecture.</a>
%e A274123 Before counting gaps, all gaps are zero, so the first pass happens after the first prime, 2. Up to and including 113, a gap of 2 occurs at least as often as any other gap. At prime 113, the gaps 2 and 4 are the most frequent (both occur 10 times). After 127, the next prime after 113, there is a gap of 4. So at the prime 127, the gap 4 has occurs the most of all gaps. This was not the case at the prime previous to 127 (the prime 113). Therefore, 127 is in the sequence.
%o A274123 (PARI) \\ See link by name "PARI program" for an extended version with comments.
%o A274123 upto(n) = {my(gapcount=List(),passes=List(),gmax = 0,imax = 0);
%o A274123 n=max(n,3); forprime(i=3, n, g = nextprime(i+1) - i; for(i = #gapcount+1, g\2, listput(gapcount,0)); gapcount[g\2]++; if(gapcount[g\2] > gmax,gmax = gapcount[g\2];if(imax!=g\2,listput(passes,i);imax=g\2)));passes[1]=2; passes} \\ _David A. Corneth_, Jun 28 2016
%Y A274123 Cf. A274121, A274122.
%K A274123 nonn,more
%O A274123 1,1
%A A274123 _David A. Corneth_, Jun 10 2016