

A023186


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


23



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
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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

Ken Takusagawa, Table of n, a(n) for n = 1..40


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

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 0; q = 2; i = 0; Do[r = NextPrim[q]; m = Min[r  q, q  p]; If[m > i, Print[q]; i = m]; p = q; q = r, {n, 1, 152382000}]


CROSSREFS

Related sequences: A023186A023188, A046929A046931, A051650, A051652, A051697A051702, A051728A051730, 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



