A023186 Lonely (or isolated) primes: increasing distance to nearest prime.

2, 5, 23, 53, 211, 1847, 2179, 3967, 16033, 24281, 38501, 58831, 203713, 206699, 413353, 1272749, 2198981, 5102953, 10938023, 12623189, 72546283, 142414669, 162821917, 163710121, 325737821, 1131241763, 1791752797, 3173306951, 4841337887, 6021542119, 6807940367, 7174208683, 8835528511, 11179888193, 15318488291, 26329105043, 31587561361, 45241670743

Offset

1,1

Comments

Erdős and Suranyi call these reclusive primes and prove that there are an infinite number of them. They define these primes to be between two primes. Hence their first term would be 3 instead of 2. Record values in A120937. - T. D. Noe, Jul 21 2006

References

Paul Erdős and Janos Suranyi, Topics in the theory of numbers, Springer, 2003.

Links

Dmitry Petukhov, Table of n, a(n) for n = 1..56 (first 40 terms from Ken Takusagawa, terms 41..52 from Giovanni Resta)

Example

The nearest prime to 23 is 4 units away, larger than any previous prime, so 23 is in the sequence.
The prime a(4) = A120937(3) = 53 is at distance 2*3 = 6 from its neighbors {47, 59}. The prime a(5) = A120937(4) = A120937(5) = A120937(6) = 211 is at distance 2*6 = 12 from its neighbors {199, 223}. Sequence A120937 requires the terms to have 2 neighbors, therefore its first term is 3 and not 2. - M. F. Hasler, Dec 28 2015

Maple

P:=proc(q) local a, b, k, n; print(2); k:=0;
for n from 3 to q do a:=ithprime(n)-prevprime(ithprime(n));
b:=nextprime(ithprime(n))-ithprime(n);
if a>b then if k<b then k:=b; print(ithprime(n)); fi;
else if k<a then k:=a; print(ithprime(n)); fi; fi;
od; end: P(10^20); # Paolo P. Lava, Jun 18 2014

Mathematica

p = 0; q = 2; i = 0; Do[r = NextPrime[q]; m = Min[r - q, q - p]; If[m > i, Print[q]; i = m]; p = q; q = r, {n, 1, 152382000}]
Join[{2}, DeleteDuplicates[{#[[2]], Min[Differences[#]]}&/@Partition[Prime[ Range[ 2, 10^6]], 3, 1], GreaterEqual[ #1[[2]], #2[[2]]]&][[;; , 1]]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Aug 31 2023 *)

Crossrefs

Related sequences: A023186, A023187, A023188, A046929, A046930, A046931, A051650, A051652, A051697-A051702, A051728, A051729, A051730, A102723.

The distances are in A023187.

Sequence in context: A156314 A308055 A173396 * A023188 A106858 A290887

Adjacent sequences: A023183 A023184 A023185 * A023187 A023188 A023189

Keyword

nonn,nice

Author

David W. Wilson

Extensions

More terms from Jud McCranie, Jun 16 2000

More terms from T. D. Noe, Jul 21 2006

Status

approved