A000230 a(0)=2; for n>=1, a(n) = smallest prime p such that there is a gap of exactly 2n between p and next prime, or -1 if no such prime exists. (Formerly M2685 N1075)

2, 3, 7, 23, 89, 139, 199, 113, 1831, 523, 887, 1129, 1669, 2477, 2971, 4297, 5591, 1327, 9551, 30593, 19333, 16141, 15683, 81463, 28229, 31907, 19609, 35617, 82073, 44293, 43331, 34061, 89689, 162143, 134513, 173359, 31397, 404597, 212701, 188029, 542603, 265621, 461717, 155921, 544279, 404851, 927869, 1100977, 360653, 604073

Offset

0,1

Comments

p + 1 = A045881(n) starts the smallest run of exactly 2n-1 successive composite numbers. - Lekraj Beedassy, Apr 23 2010

Weintraub gives upper bounds on a(252), a(255), a(264), a(273), and a(327) based on a search from 1.1 * 10^16 to 1.1 * 10^16 + 1.5 * 10^9, probably performed on a 1970s microcomputer. - Charles R Greathouse IV, Aug 26 2022

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Hugo Pfoertner, Table of n, a(n) for n = 0..672, extracted from T. Olivera e Silva's webpage.

A. Booker, The Nth Prime Page

L. J. Lander and T. R. Parkin, On the first appearance of prime differences, Math. Comp., 21 (1967), 483-488.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Tomás Oliveira e Silva, Gaps between consecutive primes

J. Thonnard, Les nombres premiers (Primality check; Closest next prime; Factorizer)

Sol Weintraub, A large prime gap, Mathematics of Computation Vol. 36, No. 153 (Jan 1981), p. 279.

J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.

Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, Volume 179 (2014), Issue 3, pp. 1121-1174.

Index entries for primes, gaps between

Formula

a(n) = A000040(A038664(n)). - Lekraj Beedassy, Sep 09 2006

Example

The following table, based on a very much larger table in the web page of Tomás Oliveira e Silva (see link) shows, for each gap g, P(g) = the smallest prime such that P(g)+g is the smallest prime number larger than P(g);
* marks a record-holder: g is a record-holder if P(g') > P(g) for all (even) g' > g, i.e., if all prime gaps are smaller than g for all primes smaller than P(g); P(g) is a record-holder if P(g') < P(g) for all (even) g' < g.
This table gives rise to many sequences: P(g) is A000230, the present sequence; P(g)* is A133430; the positions of the *'s in the P(g) column give A100180, A133430; g* is A005250; P(g*) is A002386; etc.
   -----
   g P(g)
   -----
   1* 2*
   2* 3*
   4* 7*
   6* 23*
   8* 89*
   10 139*
   12 199*
   14* 113
   16 1831*
   18* 523
   20* 887
   22* 1129
   24 1669
   26 2477*
   28 2971*
   30 4297*
   32 5591*
   34* 1327
   36* 9551*
   ........
The first time a gap of 4 occurs between primes is between 7 and 11, so a(2)=7 and A001632(2)=11.

Mathematica

Join[{2}, With[{pr = Partition[Prime[Range[86000]], 2, 1]}, Transpose[ Flatten[ Table[Select[pr, #[[2]] - #[[1]] == 2n &, 1], {n, 50}], 1]][[1]]]] (* Harvey P. Dale, Apr 20 2012 *)

Prog

(PARI) a(n)=my(p=2); forprime(q=3, , if(q-p==2*n, return(p)); p=q) \\ Charles R Greathouse IV, Nov 20 2012
(Perl) use ntheory ":all"; my($l, $i, @g)=(2, 0); forprimes { $g[($_-$l) >> 1] //= $l;  while (defined $g[$i]) { print "$i $g[$i]\n"; $i++; }  $l=$_; } 1e10; # Dana Jacobsen, Mar 29 2019
(Python)
import numpy
from sympy import sieve as prime
aupto = 50
A000230 = np.zeros(aupto+1, dtype=object)
A000230[0], it = 2, 2
while all(A000230) == 0:
    gap = (prime[it+1] - prime[it]) // 2
    if gap <= aupto and A000230[gap] == 0: A000230[gap] = prime[it]
    it += 1
print(list(A000230)) # Karl-Heinz Hofmann, Jun 07 2023

Crossrefs

A001632(n) = 2n + a(n) = nextprime(a(n)).

Cf. A001223, A002386, A005250, A045881, A038664.

Cf. A100964 (least prime number that begins a prime gap of at least 2n).

Cf. also A133429 (records), A133430, A100180, A226657, A229021, A229028, A229030, A229033, A229034.

Sequence in context: A163834 A335366 A002386 * A256454 A133429 A087770

Adjacent sequences: A000227 A000228 A000229 * A000231 A000232 A000233

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

a(29)-a(37) from Jud McCranie, Dec 11 1999

a(38)-a(49) from Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 11 2002

"or -1 if ..." added to definition at the suggestion of Alexander Wajnberg by N. J. A. Sloane, Feb 02 2020

Status

approved