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

Author

David A. Corneth, Jun 10 2016

Keywords

• nonn

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").

Examples

			(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.
		

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.

Crossrefs

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

Programs

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

Formula

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

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

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