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A159559 Lexicographically first strictly increasing sequence starting a(2) = 3 with the property that a(n) is prime if and only if n is prime.

%I A159559 #33 Mar 30 2020 20:59:13
%S A159559 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 A159559 40,42,44,47,48,53,54,55,56,57,58,59,60,62,63,67,68,71,72,74,75,79,80,
%U A159559 81,82,84,85,89,90,91,92,93,94,97,98,101,102,104,105,106,108,109,110,111
%N A159559 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 A159559 a(n) is prime iff n is prime.
%H A159559 Charles R Greathouse IV, <a href="/A159559/b159559.txt">Table of n, a(n) for n = 2..10000</a>
%H A159559 Vladimir 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 A159559 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 A159559 For n = 6, since n is composite, a(6) is the smallest composite number greater than a(6-1) = a(5) = 7, so a(6) = 8. For n = 11, since n is prime, a(11) is the smallest prime number greater than a(11-1) = a(10) = 15, so a(12) = 17. - _Michael B. Porter_, Sep 04 2016
%p A159559 A159559 := proc(n) option remember; if n = 2 then 3; else for a from procname(n-1)+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 A159559 a[2] = 3;
%t A159559 a[n_] := a[n] = If[PrimeQ[n], NextPrime[a[n-1]], NestWhile[#+1&, a[n-1]+1, PrimeQ]];
%t A159559 Map[a, Range[2, 100]] (* _Peter J. C. Moses_, Sep 19 2013 *)
%o A159559 (PARI) nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n),n+1,n)
%o A159559 first(n)=my(v=vector(n)); v[2]=3; for(k=3,n, v[k]=if(isprime(k),nextprime(v[k-1]+1), nextcomposite(v[k-1]+1))); v[2..n] \\ _Charles R Greathouse IV_, Sep 21 2016
%Y A159559 Cf. A159698, A229019, A229132.
%K A159559 nonn
%O A159559 2,1
%A A159559 _Vladimir Shevelev_, Apr 15 2009, May 04 2009
%E A159559 More terms from _R. J. Mathar_, Jul 28 2009