A192584 -id:A192584 - cp's OEIS frontend

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.

2, 5, 11, 23, 47

Offset: 1

Clark Kimberling, Jul 04 2011

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":
A192580:  f(x,y)=xy+1 and start={2}
A192581:  f(x,y)=xy+1 and start={2,4}
A192582:  f(x,y)=xy+1 and start={2,4,6}
A192583:  f(x,y)=xy+1 and start={2,4,6,8}
A192584:  f(x,y)=xy+1 and start={2,4,6,8,10}
For other choices of the function f(x,y) and start, see A192585-A192598.
A192580 consists of only 5 terms, A192581 of 7 terms, and A192582 of 28,...; what can be said about the sequence (5,7,28,...)?
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

			2 is in the sequence by decree.
The generated numbers are 5=2*2+1, 11=2*5+1, 23=2*11+1, 47=2*23+1.

Wikipedia, Cunningham chain

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

start = {2}; primes = Table[Prime[n], {n, 1, 10000}];
f[x_, y_] := If[MemberQ[primes, x*y + 1], x*y + 1]
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 = FixedPoint[b, start]  (* A192580 *)

A192585 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, 5, 8, 11, and 14 are in S.

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 23, 29, 41, 47, 59, 71, 83, 89, 113, 137, 167, 179, 197, 227, 233, 239, 359, 467, 479, 569, 659, 719, 827, 1097, 1163, 1319, 1433, 1439, 1583, 1913, 2339, 2879, 3167, 3347, 3833, 4679, 5273, 9227, 10067, 11579, 15359, 18713, 20063

Offset: 1

Clark Kimberling, Jul 05 2011

Last term is a(70) = 15785183. - Giovanni Resta, Mar 21 2013

Giovanni Resta, Table of n, a(n) for n = 1..70 (full sequence)

Cf. A192476, A192580, A192584.

start = {2, 5, 8, 11, 14}; seq = {}; new = start; While[new != {}, seq = Union[seq, new]; fresh = new; new = {}; Do[If[PrimeQ[u = x*y + 1], If[! MemberQ[seq, u], AppendTo[new, u]]], {x, seq}, {y, fresh}]]; seq (* Giovanni Resta, Mar 21 2013 *)

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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.

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