A102723 -id:A102723 - cp's OEIS frontend

A007510 Single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime.

2, 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Almost all primes are a member of this sequence by Brun's theorem.

A010051(a(n))*(1-A164292(a(n))) = 0; complement of A001097 with respect to A000040. - Reinhard Zumkeller, Mar 31 2010

			All primes congruent to 7 mod 15 are members, except for 7. All terms of A102723 are members, except for 5. - _Jonathan Sondow_, Oct 27 2017

Richard L. Francis, "Isolated Primes", J. Rec. Math., 11 (1978), 17-22.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Jens Kruse Andersen, Paul Underwood and Pierre Cami, Chen prime with 70301 digits, digest of 3 messages in primeform Yahoo group, Oct 7, 2005.
Jens Kruse Andersen, Yahoo Primeform Group Message 6481 dd. Oct 7, 2005, reconstruction in html.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Wikipedia, Isolated prime

Cf. A083370, A124582, A134099, A134100, A134101, A010051, A102723.

import Data.List (elemIndices)
a007510 n = a007510_list !! (n-1)
a007510_list = map (+ 1) $ elemIndices (0, 1, 0) $
zip3 (drop 2 a010051_list) a010051_list (0 : 0 : a010051_list)
-- Reinhard Zumkeller, Sep 16 2014

[p: p in PrimesUpTo(1000)| not IsPrime(p-2) and not IsPrime(p+2)]; // Vincenzo Librandi, Jun 20 2014

with(numtheory): for i from 1 to 150 do p:=ithprime(i): if(not isprime(p+2) and not isprime(p-2)) then printf("%d, ",p) fi od: # Pab Ter
isA007510 := proc(n) isprime(n) and not isprime(n+2) and not isprime(n-2) ; simplify(%) ; end proc:
A007510 := proc(n) if n = 1 then 2; else for a from procname(n-1)+1 do if isA007510(a) then return a; end if; end do; end if; end proc: # R. J. Mathar, Apr 26 2010

Transpose[Select[Partition[Prime[Range[100]], 3, 1], #[[2]] - #[[1]] != 2 && #[[3]] - #[[2]] != 2 &]][[2]] (* Harvey P. Dale, Mar 01 2001 *)
Select[Prime[Range[4,100]],!PrimeQ[ #-2]&&!PrimeQ[ #+2]&] (* Zak Seidov, May 07 2007 *)
Select[Prime[Range[150]],NoneTrue[#+{2,-2},PrimeQ]&] (* Harvey P. Dale, Dec 26 2022 *)

forprime(x=2,1000,if(!isprime(x-2)&&!isprime(x+2),print(x))) \\ Zak Seidov, Mar 23 2009

list(lim)=my(v=List([2]),p=3,q=5); forprime(r=7,lim, if(q-p>2 && r-q>2, listput(v,q)); p=q; q=r); p=precprime(lim); if(p<=lim && p-precprime(p-2)>2 && nextprime(p+2)-p>2, listput(v,p)); Vec(v) \\ Charles R Greathouse IV, Aug 21 2017

from sympy import nextprime
def aupto(limit):
  n, p, q = 1, 2, 3
  alst, non_twins, twins = [], [2], [3]
  while True:
    p, q = q, nextprime(q)
    if q - p == 2:
      if p != twins[-1]: twins.append(p)
      twins.append(q)
    else:
      if p != twins[-1]: non_twins.append(p)
    if q > limit: return non_twins
print(aupto(563)) # Michael S. Branicky, Feb 23 2021

10 'primes using counters 20 N=3:print "2 ";:print "3 ";:C=2 30 A=3:S=sqrt(N) 40 B=N\A 50 if B*A=N then 55 55 Q=N+2:R=N-2: if Q<>prmdiv(Q) and N=prmdiv(N) and R<>prmdiv(R) then print Q;N;R;"-";:stop:else N=N+2:goto 30 60 A=A+2 70 if A<=sqrt(N) then 40:stop 81 C=C+1 100 N=N+2:goto 30 ' Enoch Haga, Oct 08 2007

A010051(a(n)-2) + A010051(a(n)+2) = 0, n > 2. - Reinhard Zumkeller, Sep 16 2014
a(n) = prime(A176656(n)). - R. J. Mathar, Feb 19 2017
a(n) ~ n log n. - Charles R Greathouse IV, Aug 21 2017

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 11 2005

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

Original entry on oeis.org

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

David W. Wilson

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

			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

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

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

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

The distances are in A023187.

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 *)

More terms from Jud McCranie, Jun 16 2000

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

A103709 Smallest prime p such that both p +/- 2n are primes closest to p, or zero if no such prime exists.

Original entry on oeis.org

5, 0, 53, 0, 0, 211, 0, 0, 20201, 0, 0, 16787, 0, 0, 69623, 0, 0, 255803, 0, 0, 247141, 0, 0, 3565979, 0, 0, 6314447, 0, 0, 4911311, 0, 0, 12012743, 0, 0, 23346809, 0, 0, 43607429, 0, 0, 34346287, 0, 0, 36598607, 0, 0, 51042053, 0, 0, 460475569, 0, 0

Offset: 1

Zak Seidov, Feb 12 2005

nonn

Such triples of primes occur only for n divisible by 3 (except for the first term with n=1).

			a(36)=23346809 because 23346809-72, 23346809 and 23346809+72 are three successive primes and 23346809 is the least such prime.

Cf. A054342, A102723, A023186.

p-2n, p and p+2n are three successive primes and p is the least such prime.

More terms from Robert G. Wilson v, Feb 22 2005

Definition, Formula, and Example clarified by Jonathan Sondow, Oct 27 2017

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A007510 Single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime.

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

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A103709 Smallest prime p such that both p +/- 2n are primes closest to p, or zero if no such prime exists.

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