cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 22 results. Next

A192583 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, and 8 are in S.

Original entry on oeis.org

2, 4, 5, 6, 8, 11, 13, 17, 23, 31, 37, 41, 47, 53, 67, 79, 83, 89, 103, 107, 137, 139, 149, 167, 179, 223, 269, 283, 317, 359, 499, 557, 619, 643, 719, 823, 857, 1097, 1193, 1433, 1439, 1699, 1997, 2153, 2477, 2879, 3343, 4457, 6857, 7159, 8599, 12919, 41143
Offset: 1

Views

Author

Clark Kimberling, Jul 04 2011

Keywords

Comments

See the discussion at A192580.

Crossrefs

Cf. A192476.

Programs

  • Mathematica
    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". *)

A084435 a(1) = 2, then smallest prime of the form 2^k*a(n-1) + 1.

Original entry on oeis.org

2, 3, 7, 29, 59, 1889, 3779, 7559, 4058207223809, 32465657790473, 4462046030502692971872257, 9582170887127842377060195852353537
Offset: 1

Views

Author

Amarnath Murthy, Jun 03 2003

Keywords

Comments

This sequence also is generated when the initial term is 1. It is unclear if the sequence is finite or infinite. - Bob Selcoe, Oct 09 2015

Examples

			a(3)=7 because 3*2+1=7 is prime;
a(4)=29 because 7*2+1=15 is not prime, 7*4+1=29 is prime.

References

  • Donald E. Knuth, The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, problem 39, page 76.

Links

Crossrefs

Cf. A113767, A192580.

Programs

  • Mathematica
    f[s_List] := Block[{k = 0, p = s[[-1]]}, While[q = 2^k*p + 1; !PrimeQ[ q], k++]; Append[s, q]]; s = {2}; Nest[f, s, 16] (* Robert G. Wilson v, Mar 11 2015 *)
  • PARI
    lista(nn) = {a = 2; print1(a, ", "); for (n=1, nn, k=0; while (!isprime(2^k*a+1), k++); a = 2^k*a+1; print1(a, ", "););} \\ Michel Marcus, Mar 18 2015

A192582 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, and 6 are in S.

Original entry on oeis.org

2, 4, 5, 6, 11, 13, 17, 23, 31, 37, 47, 53, 67, 79, 103, 107, 139, 149, 223, 269, 283, 317, 557, 619, 643, 1699, 2477, 3343
Offset: 1

Views

Author

Clark Kimberling, Jul 04 2011

Keywords

Comments

See the discussion at A192580.

Crossrefs

Cf. A192476.

Programs

  • Mathematica
    start = {2, 4, 6}; 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]  (* A192582 *)

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

Views

Author

Clark Kimberling, Jul 04 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A192476, A192580, A192583.

Programs

  • Mathematica
    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 *)

Extensions

Corrected by Giovanni Resta, Mar 21 2013

A192586 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 and 4 are in S.

Original entry on oeis.org

2, 3, 4, 5, 7, 11, 13, 19, 37, 43, 73
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2011

Keywords

Comments

See the discussions at A192476 and A192580.

Crossrefs

Cf. A192476, A192580, A192587, A192588.

Programs

  • Mathematica
    start = {2, 4}; 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]  (* A192586 *)

A192587 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, and 6 are in S.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 67, 73, 101, 113, 137, 163, 173, 257, 401, 547, 677, 691, 821, 977, 1093, 1381, 2707, 3907, 5413, 5861
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2011

Keywords

Comments

See the discussions at A192476 and A192580.

Crossrefs

Cf. A192476, A192580, A192586, A192588.

Programs

  • Mathematica
    start = {2, 4, 6}; 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]   (* A192587 *)

A192588 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, and 8 are in S.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 61, 67, 73, 101, 103, 113, 137, 151, 163, 173, 257, 281, 401, 487, 547, 617, 677, 691, 821, 823, 977, 1093, 1123, 1303, 1381, 2467, 2707, 3701, 3907, 4933, 4937, 5413, 5527, 5861, 6737, 7817
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2011

Keywords

Comments

See the discussions at A192476 and A192580.
Last term is a(61) = 62533. - Giovanni Resta, Mar 21 2013

Links

Crossrefs

Cf. A192476, A192580, A192586, A192587.

Programs

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

A192592 Monotonic ordering of set S generated by these rules: if x and y are in S and 3x+2y is a prime, then 3x+2y is in S, and 1 is in S.

Original entry on oeis.org

1, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2011

Keywords

Comments

See the discussions at A192476 and A192580.

Links

Crossrefs

Cf. A192476, A192580, A192593.

Programs

  • Mathematica
    start = {1}; primes = Table[Prime[n], {n, 1, 10000}];
f[x_, y_] := If[MemberQ[primes, 2 x + 3 y], 2 x + 3 y]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1,
         Length[w]}]]]], # < 1000 &]];
t = FixedPoint[b, start]     (* A192592 *)
PrimePi[t]    (* A192593 *)

A192598 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2+2y^2 is a prime, then x^2+2y^2 is in S, and 1 is in S.

Original entry on oeis.org

1, 3, 11, 19, 139, 251, 379
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2011

Keywords

Comments

See the discussions at A192476 and A192580. The start-set for A192598 is {1}. For results using start-sets {1,2}, and {1,2,4}, see A192612 and A192613.

Crossrefs

Cf. A192476, A192580, A192612, A192613.

Programs

  • Mathematica
    start = {1}; primes = Table[Prime[n], {n, 1, 20000}];
f[x_, y_] := If[MemberQ[primes, x^2 + 2 y^2], x^2 + 2 y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1,
         Length[w]}]]]], # < 30000 &]];
t = FixedPoint[b, start]  (* A192598 *)

A192612 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2+2y^2 is a prime, then x^2+2y^2 is in S, and 1 and 2 are in S.

Original entry on oeis.org

1, 2, 3, 11, 17, 19, 139, 251, 307, 379, 587
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2011

Keywords

Comments

See the discussions at A192476 and A192580. The start-set for A192612 is {1,2}. For results using start-sets {1}, and {1,2,4}, see A192598 and A192613.

Crossrefs

Cf. A192476, A192580, A192613.

Programs

  • Mathematica
    start = {1, 2}; primes = Table[Prime[n], {n, 1, 20000}];
f[x_, y_] := If[MemberQ[primes, x^2 + 2 y^2], x^2 + 2 y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1,
         Length[w]}]]]], # < 30000 &]];
t = FixedPoint[b, start]  (* A192612 *)
Showing 1-10 of 22 results. Next

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