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

Author

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

Keywords

• nonn
• easy
• nice

Comments

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

Examples

			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
		

References

• 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).

Links

• 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

Crossrefs

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

Programs

Haskell

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

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Python

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

UBASIC

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

Formula

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

Extensions

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