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.

3, 5, 6, 7, 8, 11, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 32, 33, 37, 38, 39, 40, 42, 44, 47, 48, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 67, 68, 71, 72, 74, 75, 79, 80, 81, 82, 84, 85, 89, 90, 91, 92, 93, 94, 97, 98, 101, 102, 104, 105, 106, 108, 109, 110, 111

Offset: 2

Author

Vladimir Shevelev, Apr 15 2009, May 04 2009

Keywords

• nonn

Comments

a(n) is prime iff n is prime.

Examples

			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
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
• Vladimir Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009.

Crossrefs

Cf. A159698, A229019, A229132.

Programs

Maple

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

Mathematica

a[2] = 3;
a[n_] := a[n] = If[PrimeQ[n], NextPrime[a[n-1]], NestWhile[#+1&, a[n-1]+1, PrimeQ]];
Map[a, Range[2, 100]] (* Peter J. C. Moses, Sep 19 2013 *)

PARI

nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n),n+1,n)
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

Formula

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}.

Extensions

More terms from R. J. Mathar, Jul 28 2009