A014320 The next new gap between successive primes.

1, 2, 4, 6, 8, 14, 10, 12, 18, 20, 22, 34, 24, 16, 26, 28, 30, 32, 36, 44, 42, 40, 52, 48, 38, 72, 50, 62, 54, 60, 58, 46, 56, 64, 68, 86, 66, 70, 78, 76, 82, 96, 112, 100, 74, 90, 84, 114, 80, 88, 98, 92, 106, 94, 118, 132, 104, 102, 110, 126, 120, 148, 108, 122, 138

Offset: 1

Author

Hynek Mlcousek (hynek(AT)dior.ics.muni.cz)

Keywords

• nonn

Comments

Prime differences A001223 in natural order with duplicates removed. - Reinhard Zumkeller, Apr 03 2015

Conjecture: a(n) = O(n). See arXiv:2002.02115 for discussion. - Alexei Kourbatov, Jun 04 2020

Examples

			The first two primes are 2 and 3, and the first prime gap is 3 - 2 = 1; so a(1) = 1. The next prime is 5, and the next gap is 5 - 3 = 2; this gap size has not occurred before, so a(2) = 2. The next prime is 7, and the next gap is 7 - 5 = 2; the gap size 2 has already occurred before, so nothing is added to the sequence.
		

Links

• Brian Kehrig, Table of n, a(n) for n = 1..754 (terms 1..120 from Reinhard Zumkeller, 121..745 from Alexei Kourbatov).
• Alexei Kourbatov and Marek Wolf, On the first occurrences of gaps between primes in a residue class, arXiv preprint arXiv:2002.02115 [math.NT], 2020.

Crossrefs

Cf. A000101, A001223, A002386, A005250, A330853, A334543, A335366, A335367.

Programs

Haskell

import Data.List (nub)
a014320 n = a014320_list !! (n-1)
a014320_list = nub $ a001223_list
-- Reinhard Zumkeller, Apr 03 2015

Mathematica

max = 300000; allGaps = Transpose[ {gaps = Differences[ Prime[ Range[max]]], Range[ Length[gaps]]}]; equalGaps = Split[ Sort[ allGaps, #1[[1]] < #2[[1]] & ], #1[[1]] == #2[[1]] & ]; firstGaps = ((Sort[#1, #1[[1]] < #2[[1]] & ] & ) /@ equalGaps)[[All, 1]]; Sort[ firstGaps, #1[[2]] < #2[[2]] & ][[All, 1]] (* Jean-François Alcover, Oct 21 2011 *)
DeleteDuplicates[Differences[Prime[Range[10000]]]] (* Alonso del Arte, Jun 05 2020 *)

PARI

my(isFirstOcc=vector(9999, j, 1), s=2); forprime(p=3, 1e8, my(g=p-s); if(isFirstOcc[g], print1(g, ", "); isFirstOcc[g]=0); s=p) \\ Alexei Kourbatov, Jun 03 2020

Scala

val prime: LazyList[Int] = 2 #:: LazyList.from(3).filter(i => prime.takeWhile {
   j => j * j <= i
}.forall {
   k => i % k != 0
})
val primes = prime.take(1000).toList
primes.zip(primes.tail).map(p => p.2 - p._1).distinct // _Alonso del Arte, Jun 04 2020

Formula

a(n) = A335367(n) - A335366(n). - Alexei Kourbatov, Jun 04 2020

a(n) = 2*A014321(n-1) for n >= 2. - Robert Israel, May 27 2024

Extensions

More terms from Sascha Kurz, Mar 24 2002