A274121 The gap prime(n+1) - prime(n) occurs for the a(n)-th time.

1, 1, 2, 1, 3, 2, 4, 3, 1, 5, 2, 4, 6, 5, 3, 4, 7, 5, 6, 8, 6, 7, 7, 1, 8, 9, 9, 10, 10, 1, 11, 8, 11, 1, 12, 9, 10, 12, 11, 12, 13, 2, 14, 13, 15, 1, 2, 14, 16, 15, 13, 17, 3, 14, 15, 16, 18, 17, 16, 19, 4, 2, 17, 20, 18, 3, 18, 5, 21, 19, 19, 2, 20, 21, 20, 22, 3, 21, 4, 6, 22, 7, 23, 23, 22, 24, 5, 23, 24, 24, 3, 6

Offset

1,3

Comments

Terms of this sequence grow without bound; any even number occurs in this sequence. Zhang proved that there are infinitely many primes 4680 apart from each other (see link "Bounded gaps between primes").

For a conjectured count of gap n below x, see link Polignac's conjecture.

Polignac's conjecture states that "For any positive even number n, there are infinitely many prime gaps of size n.". By this conjecture, every positive apppears infinitely many times in this sequence (see link "Polignac's conjecture").

Links

David A. Corneth, Table of n, a(n) for n = 1..100021 (all terms up to 1300000)

David A. Corneth, PARI program

PolyMath, Bounded gaps between primes.

Wikipedia, Polignac's conjecture.

Formula

a(primepi(A000230(n))) = 1.

a(primepi(A001359(n))) = n.

a(primepi(A029710(n))) = n.

Example

(p, g) denotes a prime p and the gap up to the next prime. So p + g is the next prime after p. These pairs start (2, 1), (3, 2), (5, 2), (7, 4), (11, 2). From here we see that:
- the gap after the first prime, 1 occurs for the first time, so a(1) = 1.
- the gap after the second prime, 2, occurs for the first time, so a(2) = 1.
- the gap after the third prime, 2, occurs for the second time, so a(3) = 2.
- the gap after the fourth prime, 4, occurs for the first time, so a(4) = 1.
- the gap after the fifth prime, 2, occurs for the third time, so a(5) = 3.

Prog

(PARI) \\ See link by name "PARI program" for an extended version with comments.
upto(n) = {my(gapcount=List(), freqgap = List([1])); 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]++; listput(freqgap, gapcount[g\2])); freqgap} \\ David A. Corneth, Jun 28 2016

Crossrefs

Cf. A000230, A005597, A001359, A029710, A274122, A274123.

Sequence in context: A282744 A195310 A051282 * A052306 A322886 A320642

Adjacent sequences: A274118 A274119 A274120 * A274122 A274123 A274124

Keyword

nonn

Author

David A. Corneth, Jun 10 2016

Status

approved