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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.
23

%I #26 Oct 29 2015 07:15:50

%S 2,5,11,23,47

%N 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 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: f(x,y)=xy+1 and start={2}

%C A192581: f(x,y)=xy+1 and start={2,4}

%C A192582: f(x,y)=xy+1 and start={2,4,6}

%C A192583: f(x,y)=xy+1 and start={2,4,6,8}

%C A192584: f(x,y)=xy+1 and start={2,4,6,8,10}

%C For other choices of the function f(x,y) and start, see A192585-A192598.

%C 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cunningham_chain">Cunningham chain</a>

%e 2 is in the sequence by decree.

%e The generated numbers are 5=2*2+1, 11=2*5+1, 23=2*11+1, 47=2*23+1.

%t start = {2}; primes = Table[Prime[n], {n, 1, 10000}];

%t f[x_, y_] := If[MemberQ[primes, x*y + 1], x*y + 1]

%t 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 t = FixedPoint[b, start] (* A192580 *)

%Y Cf. A005384, A192476, A192581, A192582, A192583, A192584.

%K nonn,fini,full

%O 1,1

%A _Clark Kimberling_, Jul 04 2011