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 10 results.

A091514 Primes of the form (2^n + 1)^2 - 2 = 4^n + 2^(n+1) - 1.

Original entry on oeis.org

2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207, 5070602400912922109586440191999
Offset: 1

Views

Author

Eric W. Weisstein, Jan 17 2004

Keywords

Comments

Cletus Emmanuel calls these "Kynea primes".

Links

Crossrefs

Cf. A093069 (numbers of the form (2^n + 1)^2 - 2).
Cf. A091513 (indices n such that (2^n + 1)^2 - 2 is prime).

Programs

Formula

a(n) = (2^A091513(n) + 1)^2 - 2.

Extensions

Edited by Ray Chandler, Nov 15 2004
First term (2) added by Vincenzo Librandi, Dec 13 2011

A360994 Numbers k such that (2^k + 1)^3 - 2 is a semiprime.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 13, 14, 18, 27, 43, 45, 63, 76, 85, 108, 115, 119, 123, 187, 211, 215, 283, 312
Offset: 1

Views

Author

Serge Batalov, Feb 27 2023

Keywords

Comments

a(25) >= 355.
623, 674, 711, 766, 767 are also in this sequence, but their position cannot be established before finding any factor for the values corresponding to the following "blockers": 355, 511, 587, 707, 731.
1424, 1470, 1580, 1946, 2117, 2693, 3000, 3540, 4164, 7043, 9475, 10632, 15018, 19064, 27130, 28266, 28532, 46434, 58768, 103536 are some larger members of this sequence, but their position cannot be established. These produce "trivial" semiprimes where one prime is small (e.g., 3 or 5).

Links

Crossrefs

Cf. A001358, A091513, A091514, A093069, A099359, A100899, A100900, A269264, A360993.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [1..70]| IsSemiprime(s) where s is (2^n+1)^3-2];
  • Mathematica
    Select[Range[70], PrimeOmega[(2^# + 1)^3 - 2] == 2 &]
  • PARI
    isok(n) = bigomega((2^n+1)^3-2) == 2;

Formula

{ k >= 0 : A099359(k) in { A001358 } }.

A364077 Numbers k such that (12^k + 1)^2 - 2 is prime.

Original entry on oeis.org

1, 2, 8, 60, 513, 1047, 7021, 7506, 78858
Offset: 1

Views

Author

Jeppe Stig Nielsen, Jul 03 2023

Keywords

Comments

Such primes are sometimes called Kynea primes of base 12.

Links

Crossrefs

Cf. A091513 (b=2), A100902 (b=6), A100904 (b=10), A100906 (b=14), A364079 (b=18), A364081 (b=20), A100908 (b=22).
Cf. A364076.

Programs

  • PARI
    for(k=1,1200,ispseudoprime((12^k+1)^2-2)&&print1(k,", "))

A364079 Numbers k such that (18^k + 1)^2 - 2 is prime.

Original entry on oeis.org

1, 10, 21, 25, 31, 1083, 40485, 82516
Offset: 1

Views

Author

Jeppe Stig Nielsen, Jul 03 2023

Keywords

Comments

Such primes are sometimes called Kynea primes of base 18.

Links

Crossrefs

Cf. A091513 (b=2), A100902 (b=6), A100904 (b=10), A364077 (b=12), A100906 (b=14), A364081 (b=20), A100908 (b=22).
Cf. A364078.

Programs

  • PARI
    for(k=1,1200,ispseudoprime((18^k+1)^2-2)&&print1(k,", "))

A364081 Numbers k such that (20^k + 1)^2 - 2 is prime.

Original entry on oeis.org

1, 15, 44, 77, 141, 208, 304, 1169, 3359, 5050, 22431, 34935, 92990
Offset: 1

Views

Author

Jeppe Stig Nielsen, Jul 03 2023

Keywords

Comments

Such primes are sometimes called Kynea primes of base 20.

Links

Crossrefs

Cf. A091513 (b=2), A100902 (b=6), A100904 (b=10), A364077 (b=12), A100906 (b=14), A364079 (b=18), A100908 (b=22).
Cf. A364080.

Programs

  • PARI
    for(k=1,1200,ispseudoprime((20^k+1)^2-2)&&print1(k,", "))

A100496 Numbers n such that (2^n+1)^4-2 is prime.

Original entry on oeis.org

1, 7, 25, 31, 34, 271, 514, 2896, 8827, 16816, 37933
Offset: 1

Views

Author

Jonathan Vos Post, Nov 23 2004

Keywords

Comments

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019
a(12) > 60000. - Tyler Busby, Feb 12 2023

Examples

			a(1) = 1 because (2^1+1)^4 - 2 = 79 is prime and is the first such prime.

Crossrefs

Cf. A091513, A091514, A093069, A099359.
Cf. A100497, n such that (2^n+1)^4-2 is semiprime.

Programs

  • Mathematica
    Select[Range[5000], PrimeQ[(2^# + 1)^4 - 2] &]
  • PARI
    is(n)=ispseudoprime((2^n+1)^4-2) \\ Charles R Greathouse IV, Jun 13 2017

Extensions

Edited, corrected and extended by Ray Chandler and Hugo Pfoertner, Nov 26 2004
a(10)-a(11) from Tyler Busby, Feb 12 2023

A268574 Numbers k such that (2^k + 1)^2 - 2 is a semiprime.

Original entry on oeis.org

4, 6, 7, 10, 11, 14, 22, 36, 38, 39, 44, 45, 48, 49, 60, 72, 74, 75, 89, 92, 96, 99, 105, 110, 111, 113, 116, 131, 138, 143, 150, 170, 173, 182, 194, 201, 212, 234, 260, 282, 300, 317, 335, 341, 345, 383, 405
Offset: 1

Views

Author

Vincenzo Librandi, Feb 21 2016

Keywords

Comments

a(48) >= 428. - Serge Batalov, Feb 25 2023

Examples

			a(1) = 4 because 17^2 - 2 = 287 = 7*41, which is semiprime.
a(2) = 6 because 65^2 - 2 = 4223 = 41*103, which is semiprime.

Links

Crossrefs

Cf. A091513, A091514, A093069, A269264, A360993, A360994.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [1..110]| IsSemiprime(s) where s is (2^n+1)^2-2];
  • Mathematica
    Select[Range[105], PrimeOmega[(2^# + 1)^2 - 2] == 2 &]
  • PARI
    isok(n) = bigomega((2^n+1)^2-2) == 2; \\ Michel Marcus, Feb 22 2016

Extensions

a(25)-a(39) from Hugo Pfoertner, Aug 05 2019
a(40)-a(41) from chris2be8@yahoo.com, Feb 25 2023
a(42)-a(47) from Serge Batalov, Feb 26 2023

A099360 a(n) = (2^n + 1)^4 - 2.

Original entry on oeis.org

14, 79, 623, 6559, 83519, 1185919, 17850623, 276922879, 4362470399, 69257922559, 1103812890623, 17626570956799, 281749955297279, 4505799053311999, 72075187834650623, 1153062248537784319, 18447869999386460159
Offset: 0

Views

Author

Parthasarathy Nambi, Nov 16 2004

Keywords

Links

Crossrefs

Cf. A091513, A091515.

Programs

Extensions

More terms from Stefan Steinerberger, Mar 06 2006

A173888 Exactly one of (2^n-1)^2-2 and (2^n+1)^2-2 is prime.

Original entry on oeis.org

0, 1, 4, 5, 6, 7, 8, 9, 10, 17, 19, 23, 25, 32, 51, 55, 65, 87, 129, 132, 159, 171, 175, 180, 242, 315, 324, 358, 393, 435, 467, 491, 501, 507, 555, 591, 680, 786, 800, 1070, 1459, 1650, 1707, 2813, 2923, 3281, 4217, 5153, 6287, 6365, 6462, 10088, 10367, 14289
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 01 2010

Keywords

Comments

The numbers which are in A091513 or A091515, but not in both sequences. - R. J. Mathar, Mar 09 2010

Examples

			a(1)=1 because (2^1-1)^2-2=-1 is nonprime and (2^1+1)^2-2=7 is prime.

Crossrefs

Cf. A091513, A091515, A091514, A091516.

Extensions

Corrected (0 inserted, 12, 16, 18, 21 removed) and extended by R. J. Mathar, Mar 09 2010

A173903 Numbers k such that both (2^k+1)^2-2 and (2^k-1)^2-2 are prime.

Original entry on oeis.org

2, 3, 12, 15, 18, 21, 27
Offset: 1

Views

Author

Vincenzo Librandi, Mar 08 2010

Keywords

Comments

a(8) > 9394. - Max Z. Scialabba, Jan 21 2024
a(8) > 695631 using A091513 and A091515. - Michael S. Branicky, Oct 24 2024

Crossrefs

Cf. A091513, A091515.

Programs

  • Magma
    [n: n in [1..400] | IsPrime((2^n-1)^2-2) and IsPrime((2^n+1)^2-2)]
  • Mathematica
    Select[Range[3000], PrimeQ[((2^# + 1)^2 - 2)]&&PrimeQ[((2^# - 1)^2 - 2)] &] (* Vincenzo Librandi, Aug 21 2014 *)

Formula

A091513 INTERSECT A091515. - R. J. Mathar, Jul 06 2010

Extensions

Definition clarified by Jon E. Schoenfield, Jun 18 2010
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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