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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 (list; graph; refs; listen; history; text; internal format)
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
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
The distances are in A023187.
Sequence in context: A156314 A308055 A173396 * A023188 A106858 A290887
KEYWORD
nonn,nice,changed
AUTHOR
EXTENSIONS
More terms from Jud McCranie, Jun 16 2000
More terms from T. D. Noe, Jul 21 2006
STATUS
approved

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Last modified February 24 18:59 EST 2024. Contains 370307 sequences. (Running on oeis4.)