A031131 Difference between n-th prime and (n+2)-nd prime.

3, 4, 6, 6, 6, 6, 6, 10, 8, 8, 10, 6, 6, 10, 12, 8, 8, 10, 6, 8, 10, 10, 14, 12, 6, 6, 6, 6, 18, 18, 10, 8, 12, 12, 8, 12, 10, 10, 12, 8, 12, 12, 6, 6, 14, 24, 16, 6, 6, 10, 8, 12, 16, 12, 12, 8, 8, 10, 6, 12, 24, 18, 6, 6, 18, 20, 16, 12, 6, 10, 14, 14, 12, 10, 10, 14, 12, 12, 18, 12, 12, 12

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

Distance between the pair of primes adjacent to the (n+1)-st prime. - Lekraj Beedassy, Oct 01 2004 [Typo corrected by Zak Seidov, Feb 22 2009]

A031131(A261525(n)) = 2*n and A031131(m) != 2*n for m < A261525(n). - Reinhard Zumkeller, Aug 23 2015

The Polymath project 8b proved that a(n) <= 395106 infinitely often (their published paper contains the slightly weaker bound a(n) <= 398130 infinitely often). - Charles R Greathouse IV, Jul 22 2016

Examples

			a(10)=8 because the 10th prime=29 is followed by primes 31 and 37, and 37 - 29 = 8.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Research in the Mathematical Sciences 1:12 (2014).
• Polymath project, Bounded gaps between primes

Crossrefs

Sum of consecutive terms of A001223.

Cf. A031132, A031133, A031134, A122412, A122413,A046931, A000040, A031165, A031166, A031167, A031168, A031169, A031170, A031171, A031172, A261525, A052288.

Cf. A075527 (allowing 1 to be prime).

First differences of A001043.

Programs

Haskell

a031131 n = a031131_list !! (n-1)
a031131_list = zipWith (-) (drop 2 a000040_list) a000040_list
-- Reinhard Zumkeller, Dec 19 2013

Magma

[NthPrime(n+2)-NthPrime(n): n in [1..100] ]; // Vincenzo Librandi, Apr 11 2011

Maple

P:= select(isprime, [2,seq(2*i+1,i=1..1000)]):
P[3..-1] - P[1..-3]; # Robert Israel, Jan 25 2015

Mathematica

Differences[lst_]:=Drop[lst,2]-Drop[lst,-2]; Differences[Prime[Range[123]]] (* Vladimir Joseph Stephan Orlovsky, Aug 13 2009 *)
Map[#3 - #1 & @@ # &, Partition[Prime@ Range[84], 3, 1]] (* Michael De Vlieger, Dec 17 2017 *)

MuPAD

ithprime(i+2)-ithprime(i) $ i = 1..65 // Zerinvary Lajos, Feb 26 2007

PARI

a(n)=my(p=prime(n));nextprime(nextprime(p+1)+1)-p \\ Charles R Greathouse IV, Jul 01 2013

Sage

BB = primes_first_n(67)
L = []
for i in range(65):
    L.append(BB[2+i]-BB[i])
L
# Zerinvary Lajos, May 14 2007

Formula

a(n) = A001223(n) + A001223(n-1). - Lior Manor, Jan 19 2005

a(n) = A000040(n+2) - A000040(n).

a(n) = 2*A052288(n-1) for n>1. - Hugo Pfoertner, Apr 16 2025

Extensions

Corrected by T. D. Noe, Sep 11 2008

Edited by N. J. A. Sloane, Sep 18 2008, at the suggestion of T. D. Noe