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 A007510 Single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime. (Formerly M2130) 94
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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 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 EXAMPLE 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 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 *) PROG (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 (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 (Magma) [p: p in PrimesUpTo(1000)| not IsPrime(p-2) and not IsPrime(p+2)]; // Vincenzo Librandi, Jun 20 2014 (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 (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 CROSSREFS Cf. A083370, A124582, A134099, A134100, A134101, A010051, A102723. Sequence in context: A049556 A049580 A061448 * A117242 A144550 A106736 Adjacent sequences: A007507 A007508 A007509 * A007511 A007512 A007513 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane, Robert G. Wilson v EXTENSIONS More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 11 2005 STATUS approved

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Last modified February 28 12:52 EST 2024. Contains 370394 sequences. (Running on oeis4.)