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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.
23
2, 5, 11, 23, 47
OFFSET
1,1
COMMENTS
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
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Clark Kimberling, Jul 04 2011
STATUS
approved