A192583 -id:A192583 - cp's OEIS frontend

A192530 Index-list (modified) of the primes generated at A192583.

1, 2, 3, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15, 16, 19, 22, 23, 24, 27, 28, 33, 34, 35, 39, 41, 48, 57, 61, 66, 72, 95, 102, 114, 117, 128, 143, 148, 184, 196, 227, 228, 266, 302, 325, 367, 417, 471, 606, 882, 916, 1071, 1539, 4305

Offset: 1

Clark Kimberling, Jul 04 2011

Besides the generated primes 2,5,11,13,17,..., the initial numbers 4,6,8 in A192583 are represented here by index of nearest lower prime.

			A192583=(2,4,5,6,8,11,13,17,23,...).  a(1)=1 because the index of 2 is 1; a(2)=2 because, for term #2 of A192530, which is 4, the nearest prime <4 is 3, which has index 2; a(3)=3 because the index of 5 is 3.  ("Nearest prime down" for nonprimes is given by PrimePi in the Mathematica program.)

Cf. A192583.

start = {2, 4, 6, 8}; 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}]]]], # <
      10000000 &]];
t = FixedPoint[b, start]  (* A192583 *)
PrimePi[t] (* A192530 Nonprimes 4,6,8 are represented by "next prime down". *)

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.

Original entry on oeis.org

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 *)

A192584 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, 4, 6, 8, and 10 are in S.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 11, 13, 17, 23, 31, 37, 41, 47, 53, 61, 67, 79, 83, 89, 101, 103, 107, 131, 137, 139, 149, 167, 179, 223, 263, 269, 283, 311, 317, 359, 367, 499, 557, 607, 619, 643, 719, 787, 809, 823, 857, 1031, 1049, 1097, 1193, 1433, 1439, 1579, 1619

Offset: 1

Clark Kimberling, Jul 04 2011

See the discussions at A192580 and A192584. The number of terms in this finite sequence is 104. The greatest term is 15845273.

Giovanni Resta, Table of n, a(n) for n = 1..104 (full sequence)

Cf. A192476, A192580, A192583.

start = {2, 4, 6, 8, 10}; 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 *)

Corrected by Giovanni Resta, Mar 21 2013

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A192530 Index-list (modified) of the primes generated at A192583.

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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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A192584 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, 4, 6, 8, and 10 are in S.

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