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

1,1

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.

Links

Table of n, a(n) for n=1..8.

Wikipedia, Polignac's conjecture.

Example

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.

Prog

(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

Crossrefs

Cf. A274121, A274122.

Sequence in context: A139904 A167414 A065381 * A141928 A343184 A062588

Adjacent sequences: A274120 A274121 A274122 * A274124 A274125 A274126

Keyword

nonn,more

Author

David A. Corneth, Jun 10 2016

Status

approved