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%I #48 Feb 18 2024 02:02:36
%S 2,5,23,53,211,1847,2179,3967,16033,24281,38501,58831,203713,206699,
%T 413353,1272749,2198981,5102953,10938023,12623189,72546283,142414669,
%U 162821917,163710121,325737821,1131241763,1791752797,3173306951,4841337887,6021542119,6807940367,7174208683,8835528511,11179888193,15318488291,26329105043,31587561361,45241670743
%N Lonely (or isolated) primes: increasing distance to nearest prime.
%C 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
%D Paul Erdős and Janos Suranyi, Topics in the theory of numbers, Springer, 2003.
%H Dmitry Petukhov, <a href="/A023186/b023186.txt">Table of n, a(n) for n = 1..56</a> (first 40 terms from Ken Takusagawa, terms 41..52 from Giovanni Resta)
%e The nearest prime to 23 is 4 units away, larger than any previous prime, so 23 is in the sequence.
%e 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
%t 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}]
%t 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 *)
%Y Related sequences: A023186, A023187, A023188, A046929, A046930, A046931, A051650, A051652, A051697-A051702, A051728, A051729, A051730, A102723.
%Y The distances are in A023187.
%K nonn,nice
%O 1,1
%A _David W. Wilson_
%E More terms from _Jud McCranie_, Jun 16 2000
%E More terms from _T. D. Noe_, Jul 21 2006