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A192580 Monotonic ordering of set S generated by these rules: if x and y are in S and xy+1 is a prime, then xy+1 is in S, and 2 is in S.

%I A192580 #26 Oct 29 2015 07:15:50
%S A192580 2,5,11,23,47
%N A192580 Monotonic ordering of set S generated by these rules: if x and y are in S and xy+1 is a prime, then xy+1 is in S, and 2 is in S.
%C A192580 Following the discussion at A192476, the present sequence introduces a restriction: that the generated terms must be prime. A192580 is the first of an ascending chain of finite sequences, determined by the initial set called "start":
%C A192580 A192580:  f(x,y)=xy+1 and start={2}
%C A192580 A192581:  f(x,y)=xy+1 and start={2,4}
%C A192580 A192582:  f(x,y)=xy+1 and start={2,4,6}
%C A192580 A192583:  f(x,y)=xy+1 and start={2,4,6,8}
%C A192580 A192584:  f(x,y)=xy+1 and start={2,4,6,8,10}
%C A192580 For other choices of the function f(x,y) and start, see A192585-A192598.
%C A192580 A192580 consists of only 5 terms, A192581 of 7 terms, and A192582 of 28,...; what can be said about the sequence (5,7,28,...)?
%C A192580 2, 5, 11, 23, 47 is the complete Cunningham chain that begins with 2. Each term except the last is a Sophie Germain prime A005384. - _Jonathan Sondow_, Oct 28 2015
%H A192580 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cunningham_chain">Cunningham chain</a>
%e A192580 2 is in the sequence by decree.
%e A192580 The generated numbers are 5=2*2+1, 11=2*5+1, 23=2*11+1, 47=2*23+1.
%t A192580 start = {2}; primes = Table[Prime[n], {n, 1, 10000}];
%t A192580 f[x_, y_] := If[MemberQ[primes, x*y + 1], x*y + 1]
%t A192580 b[x_] := Block[{w = x}, Select[Union[Flatten[AppendTo[w, Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # < 50000 &]];
%t A192580 t = FixedPoint[b, start]  (* A192580 *)
%Y A192580 Cf. A005384, A192476, A192581, A192582, A192583, A192584.
%K A192580 nonn,fini,full
%O A192580 1,1
%A A192580 _Clark Kimberling_, Jul 04 2011