A000230 - OEIS

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)

107

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

(list; graph; refs; listen; history; text; internal format)

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