

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)


105



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
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OFFSET

0,1


COMMENTS

p + 1 = A045881(n) starts the smallest run of exactly 2n1 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

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


FORMULA



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 recordholder: g is a recordholder 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 recordholder if P(g') < P(g) for all (even) g' < g.

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

(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)
gap = (prime[it+1]  prime[it]) // 2
it += 1


CROSSREFS

A001632(n) = 2n + a(n) = nextprime(a(n)).
Cf. A100964 (least prime number that begins a prime gap of at least 2n).


KEYWORD

nonn,nice


AUTHOR



EXTENSIONS

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



