1,2

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

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.

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

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.

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 *)

(Haskell)

import Data.List (nub)

a014320 n = a014320_list !! (n-1)

a014320_list = nub $ a001223_list

-- Reinhard Zumkeller, Apr 03 2015

(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

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

Sequence in context: A387118 A072791 A058320 * A080377 A086526 A086529

Adjacent sequences: A014317 A014318 A014319 * A014321 A014322 A014323

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

More terms from Sascha Kurz, Mar 24 2002

approved