%I
%S 3,5,6,7,8,11,12,14,15,17,18,19,20,21,22,23,24,29,30,32,33,37,38,39,
%T 40,42,44,47,48,53,54,55,56,57,58,59,60,62,63,67,68,71,72,74,75,79,80,
%U 81,82,84,85,89,90,91,92,93,94,97,98,101,102,104,105,106,108,109,110,111
%N Lexicographically first strictly increasing sequence starting a(2) = 3 with the property that a(n) is prime if and only if n is prime.
%C a(n) is prime iff n is prime.
%H Charles R Greathouse IV, <a href="/A159559/b159559.txt">Table of n, a(n) for n = 2..10000</a>
%H V. Shevelev, <a href="http://arXiv.org/abs/0904.2101">Several results on sequences which are similar to the positive integers</a>, arXiv:0904.2101 [math.NT], 2009.
%F a(n+1) = min{m>a(n), m is prime}, if n+1 is prime; otherwise, a(n+1) = min{m>a(n), m is composite}.
%e For n = 6, since n is composite, a(n) is the smallest composite number greater than a(61) = a(5) = 7, so a(6) = 8. For n = 11, since n is prime, a(n) is the smallest prime number greater than a(111) = a(10) = 15, so a(12) = 17.  _Michael B. Porter_, Sep 04 2016
%p A159559 := proc(n) option remember; if n = 2 then 3; else for a from procname(n1)+1 do if isprime(n) and isprime(a) then RETURN(a) ; elif not isprime(n) and not isprime(a) then RETURN(a) ; fi; od: fi; end: seq(A159559(n),n=2..100) ; # _R. J. Mathar_, Jul 28 2009
%t a[2] = 3;
%t a[n_] := a[n] = If[PrimeQ[n], NextPrime[a[n1]], NestWhile[#+1&, a[n1]+1, PrimeQ]];
%t Map[a, Range[2, 100]] (* _Peter J. C. Moses_, Sep 19 2013 *)
%o (PARI) nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n),n+1,n)
%o first(n)=my(v=vector(n)); v[2]=3; for(k=3,n, v[k]=if(isprime(k),nextprime(v[k1]+1), nextcomposite(v[k1]+1))); v[2..n] \\ _Charles R Greathouse IV_, Sep 21 2016
%Y Cf. A159698, A229019, A229132.
%K nonn
%O 2,1
%A _Vladimir Shevelev_, Apr 15 2009, May 04 2009
%E More terms from _R. J. Mathar_, Jul 28 2009
