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.

2, 127, 149, 383, 431, 443, 487, 557

Offset: 1

Author

David A. Corneth, Jun 10 2016

Keywords

• nonn
• more

Comments

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.

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.

Examples

			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.
		

Links

• Wikipedia, Polignac's conjecture.

Crossrefs

Cf. A274121, A274122.

Programs

PARI

\\ See link by name "PARI program" for an extended version with comments.
upto(n) = {my(gapcount=List(),passes=List(),gmax = 0,imax = 0);
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