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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.

%I A000230 M2685 N1075 #156 Feb 12 2025 14:21:33
%S A000230 2,3,7,23,89,139,199,113,1831,523,887,1129,1669,2477,2971,4297,5591,
%T A000230 1327,9551,30593,19333,16141,15683,81463,28229,31907,19609,35617,
%U A000230 82073,44293,43331,34061,89689,162143,134513,173359,31397,404597,212701,188029,542603,265621,461717,155921,544279,404851,927869,1100977,360653,604073
%N 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.
%C A000230 p + 1 = A045881(n) starts the smallest run of exactly 2n-1 successive composite numbers. - _Lekraj Beedassy_, Apr 23 2010
%C A000230 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
%D A000230 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000230 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000230 Brian Kehrig, <a href="/A000230/b000230.txt">Table of n, a(n) for n = 0..721</a> (terms 0..672 from Hugo Pfoertner)
%H A000230 A. Booker, <a href="https://t5k.org/nthprime">The Nth Prime Page</a>
%H A000230 L. J. Lander and T. R. Parkin, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0230677-4">On the first appearance of prime differences</a>, Math. Comp., 21 (1967), 483-488.
%H A000230 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101]
%H A000230 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/gaps.html">Gaps between consecutive primes</a>
%H A000230 J. Thonnard, <a href="http://www.proftnj.com/calcprem.htm">Les nombres premiers (Primality check; Closest next prime; Factorizer)</a>
%H A000230 Sol Weintraub, <a href="https://doi.org/10.1090/S0025-5718-1981-0595062-1">A large prime gap</a>, Mathematics of Computation Vol. 36, No. 153 (Jan 1981), p. 279.
%H A000230 J. Young and A. Potler, <a href="http://dx.doi.org/10.1090/S0025-5718-1989-0947470-1">First occurrence prime gaps</a>, Math. Comp., 52 (1989), 221-224.
%H A000230 Yitang Zhang, <a href="https://doi.org/10.4007/annals.2014.179.3.7">Bounded gaps between primes</a>, Annals of Mathematics, Volume 179 (2014), Issue 3, pp. 1121-1174.
%H A000230 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F A000230 a(n) = A000040(A038664(n)). - _Lekraj Beedassy_, Sep 09 2006
%e A000230 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);
%e A000230 * 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.
%e A000230 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.
%e A000230    -----
%e A000230    g P(g)
%e A000230    -----
%e A000230    1* 2*
%e A000230    2* 3*
%e A000230    4* 7*
%e A000230    6* 23*
%e A000230    8* 89*
%e A000230    10 139*
%e A000230    12 199*
%e A000230    14* 113
%e A000230    16 1831*
%e A000230    18* 523
%e A000230    20* 887
%e A000230    22* 1129
%e A000230    24 1669
%e A000230    26 2477*
%e A000230    28 2971*
%e A000230    30 4297*
%e A000230    32 5591*
%e A000230    34* 1327
%e A000230    36* 9551*
%e A000230    ........
%e A000230 The first time a gap of 4 occurs between primes is between 7 and 11, so a(2)=7 and A001632(2)=11.
%t A000230 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 *)
%o A000230 (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
%o A000230 (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
%o A000230 (Python)
%o A000230 import numpy
%o A000230 from sympy import sieve as prime
%o A000230 aupto = 50
%o A000230 A000230 = np.zeros(aupto+1, dtype=object)
%o A000230 A000230[0], it = 2, 2
%o A000230 while all(A000230) == 0:
%o A000230     gap = (prime[it+1] - prime[it]) // 2
%o A000230     if gap <= aupto and A000230[gap] == 0: A000230[gap] = prime[it]
%o A000230     it += 1
%o A000230 print(list(A000230)) # _Karl-Heinz Hofmann_, Jun 07 2023
%Y A000230 A001632(n) = 2n + a(n) = nextprime(a(n)).
%Y A000230 Cf. A001223, A002386, A005250, A045881, A038664.
%Y A000230 Cf. A100964 (least prime number that begins a prime gap of at least 2n).
%Y A000230 Cf. also A133429 (records), A133430, A100180, A226657, A229021, A229028, A229030, A229033, A229034.
%K A000230 nonn,nice
%O A000230 0,1
%A A000230 _N. J. A. Sloane_
%E A000230 a(29)-a(37) from _Jud McCranie_, Dec 11 1999
%E A000230 a(38)-a(49) from Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 11 2002
%E A000230 "or -1 if ..." added to definition at the suggestion of Alexander Wajnberg by _N. J. A. Sloane_, Feb 02 2020