

A023186


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


25



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



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;


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, A051697A051702, A051728, A051729, A051730, A102723.


KEYWORD

nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



