A229028 -id:A229028 - cp's OEIS frontend

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.

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

N. J. A. Sloane

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

			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.

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).

Brian Kehrig, Table of n, a(n) for n = 0..721 (terms 0..672 from Hugo Pfoertner)
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

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.

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 *)

a(n)=my(p=2);forprime(q=3,,if(q-p==2*n,return(p));p=q) \\ Charles R Greathouse IV, Nov 20 2012

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

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

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

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

A133429 Records in A000230.

Original entry on oeis.org

2, 3, 7, 23, 89, 139, 199, 1831, 2477, 2971, 4297, 5591, 9551, 30593, 81463, 82073, 89689, 162143, 173359, 404597, 542603, 544279, 927869, 1100977, 1444309, 2238823, 5845193, 6752623, 6958667, 7621259, 10343761, 11981443, 13626257, 17983717, 49269581, 83751121

Offset: 1

N. J. A. Sloane, Nov 28 2007

nonn

N. J. A. Sloane, Table of n, a(n) for n=1..123 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Gaps between consecutive primes

Cf. A000230, A133430, A226657, A229021, A229028, A229030, A229033, A229034.

A133430 Where records occur in A000230.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 13, 14, 15, 16, 18, 19, 23, 28, 32, 33, 35, 37, 40, 44, 46, 47, 51, 54, 58, 62, 67, 70, 71, 72, 75, 78, 79, 83, 93, 97, 100, 112, 113, 114, 127, 128, 132, 139, 147, 149, 157, 158, 164, 167, 181, 184, 185, 194, 211, 218, 221, 226, 233, 235, 236, 241, 244

Offset: 1

N. J. A. Sloane, Nov 28 2007

nonn

N. J. A. Sloane, Table of n, a(n) for n=1..123 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Gaps between consecutive primes

Cf. A000230, A133429, A226657, A229021, A229028, A229030, A229033, A229034.

A226657 Smallest of the first four consecutive primes that comprise two sets of primes with difference 2*n.

Original entry on oeis.org

5, 7, 23, 389, 409, 1511, 5309, 3373, 7351, 37223, 19867, 18593, 142811, 14563, 13933, 763271, 276637, 174491, 363989, 383179, 180907, 687179, 8066923, 913589, 458069, 6358777, 2507093, 5650871, 9182389, 5256071, 10237391, 9955009, 4091393, 24374033

Offset: 1

Arkadiusz Wesolowski, Jun 14 2013

nonn

An equivalent definition of this sequence: smallest prime which gives a cluster of primes with the spacing pattern 2*n; x; 2*n, x > 0.

A229021 gives the record values. - Arkadiusz Wesolowski, Sep 11 2013

			Difference two - primes: 5, 7, 11, 13.
Difference four - primes: 7, 11, 13, 17.
Difference six - primes: 23, 29, 31, 37.

Index entries for primes, gaps between

Cf. A000230, A133429, A133430, A229021, A229028, A229030, A229033, A229034.

lst = {}; Do[a = 3; While[True, b = NextPrime[a]; If[b - a == n && NextPrime[b, 2] - NextPrime[b] == n, AppendTo[lst, a]; Break[]]; a = b], {n, 2, 68, 2}]; lst
Table[SelectFirst[Partition[Prime[Range[16*10^5]],4,1],AllTrue[{#[[2]]-#[[1]],#[[4]]- #[[3]]}, EvenQ]&&#[[2]]-#[[1]]==#[[4]]-#[[3]]==2n&],{n,35}][[All,1]] (* Harvey P. Dale, Jun 07 2022 *)

A229021 Record values in A226657.

Original entry on oeis.org

5, 7, 23, 389, 409, 1511, 5309, 7351, 37223, 142811, 763271, 8066923, 9182389, 10237391, 24374033, 70353383, 128463691, 334100083, 358453847, 610611193

Offset: 1

Arkadiusz Wesolowski, Sep 11 2013

Cf. A000230, A133429, A133430, A226657, A229028, A229030, A229033, A229034.

A229030 Smallest of the first six consecutive primes that comprise three sets of primes with difference 2*n.

Original entry on oeis.org

5, 7, 251, 683, 2017, 18679, 13499, 608131, 97213, 937127, 891997, 531359, 490283, 637171, 892321, 21954731, 5995783, 3440627, 12024413, 3697249, 2674579, 95270633, 165066283, 25091659, 465512447, 161732947, 88360297, 804346451, 286775719, 198215821

Offset: 1

Arkadiusz Wesolowski, Sep 11 2013

nonn

An equivalent definition of this sequence: smallest prime which gives a cluster of primes with the spacing pattern 2*n; x; 2*n; x; 2*n, x > 0.

A229033 gives the record values.

			Difference two - primes: 5, 7, 11, 13, 17, 19.
Difference four - primes: 7, 11, 13, 17, 19, 23.
Difference six - primes: 251, 257, 263, 269, 271, 277.

Index entries for primes, gaps between

Cf. A000230, A133429, A133430, A226657, A229021, A229028, A229033, A229034.

Table[With[{prs=Partition[Prime[Range[42000000]],6,1]},Select[prs, Union[ Take[ Differences[#1],{1,5,2}]]=={2n}&,1][[1,1]]],{n,30}] (* Harvey P. Dale, Apr 18 2014 *)

A229033 Record values in A229030.

Original entry on oeis.org

5, 7, 251, 683, 2017, 18679, 608131, 937127, 21954731, 95270633, 165066283, 465512447, 804346451

Offset: 1

Arkadiusz Wesolowski, Sep 11 2013

Cf. A000230, A133429, A133430, A226657, A229021, A229028, A229030, A229034.

A229034 Indices of records in A229030.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 16, 22, 23, 25, 28

Offset: 1

Arkadiusz Wesolowski, Sep 11 2013

Cf. A000230, A133429, A133430, A226657, A229021, A229028, A229030, A229033.

cp's OEIS Frontend

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.

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A133429 Records in A000230.

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A133430 Where records occur in A000230.

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A226657 Smallest of the first four consecutive primes that comprise two sets of primes with difference 2*n.

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A229021 Record values in A226657.

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A229030 Smallest of the first six consecutive primes that comprise three sets of primes with difference 2*n.

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A229033 Record values in A229030.

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A229034 Indices of records in A229030.

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