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
Examples
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.
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
- 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
Crossrefs
Programs
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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 *) -
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
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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
Formula
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
Comments