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.
Wikipedia, Isolated prime
FORMULA
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
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 11 2005
STATUS
approved