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

1,1

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

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

Formula

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

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

Example

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

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 *)

Prog

(MuPAD) ithprime(i+2)-ithprime(i) $ i = 1..65 // Zerinvary Lajos, Feb 26 2007
(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
(Magma) [NthPrime(n+2)-NthPrime(n): n in [1..100] ]; // Vincenzo Librandi, Apr 11 2011
(PARI) a(n)=my(p=prime(n)); nextprime(nextprime(p+1)+1)-p \\ Charles R Greathouse IV, Jul 01 2013
(Haskell)
a031131 n = a031131_list !! (n-1)
a031131_list = zipWith (-) (drop 2 a000040_list) a000040_list
-- Reinhard Zumkeller, Dec 19 2013

Crossrefs

Sum of consecutive terms of A001223.

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

Cf. A075527 (allowing 1 to be prime).

First differences of A001043.

Sequence in context: A198617 A298808 A033957 * A105321 A217032 A300305

Adjacent sequences: A031128 A031129 A031130 * A031132 A031133 A031134

Keyword

nonn

Author

Jeff Burch

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

Status

approved