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 1-9 of 9 results.

A091513 Numbers k such that (2^k + 1)^2 - 2 = 4^k + 2^(k+1) - 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 9, 12, 15, 17, 18, 21, 23, 27, 32, 51, 65, 87, 180, 242, 467, 491, 501, 507, 555, 591, 680, 800, 1070, 1650, 2813, 3281, 4217, 5153, 6287, 6365, 10088, 10367, 37035, 45873, 69312, 102435, 106380, 108888, 110615, 281621, 369581, 376050, 442052, 621443, 661478
Offset: 1

Views

Author

Eric W. Weisstein, Jan 17 2004

Keywords

Links

Crossrefs

Cf. A091514 (primes of the form (2^n + 1)^2 - 2).
Cf. A093069 (numbers of the form (2^n + 1)^2 - 2).

Programs

Formula

A093069(n) = (2^a(n) + 1)^2 - 2.

Extensions

a(41) from Eric W. Weisstein, Feb 27 2004
a(42) to a(44) from Eric W. Weisstein, Jun 05 2004
Edited by Ray Chandler, Nov 15 2004
a(46) from Cletus Emmanuel (cemmanu(AT)yahoo.com), Oct 07 2005
a(47)-a(48) from Eric W. Weisstein, Feb 10 2016 (computed by Mark Rodenkirch)
a(49)-a(50) from Eric W. Weisstein, Jun 08 2016 (computed by Mark Rodenkirch)
a(51) from Eric W. Weisstein, Jun 19 2016 (computed by Mark Rodenkirch)

A093069 a(n) = (2^n + 1)^2 - 2.

Original entry on oeis.org

7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247, 268468223, 1073807359, 4295098367, 17180131327, 68720001023, 274878955519, 1099513724927, 4398050705407, 17592194433023, 70368760954879, 281475010265087, 1125899973951487
Offset: 1

Views

Author

Eric W. Weisstein, Mar 17 2004

Keywords

Comments

Cletus Emmanuel calls these "Kynea numbers".
Difference between the smallest digitally balanced number with 2n+4 binary digits and the largest digitally balanced number with 2n+2 binary digits (see A031443): 7 = 9-2 = 1001-10, 23 = 35-12 = 100011-1100, 79 = 135-56 = 10000111-111000 etc. - Juri-Stepan Gerasimov, Jun 01 2011

Examples

			G.f. = 7*x + 23*x^2 + 79*x^3 + 287*x^4 + 1087*x^5 + 4223*x^6 + 16639*x^7 + ...

Links

Crossrefs

Cf. A091514 (primes of the form (2^n + 1)^2 - 2).
Cf. A244663.

Programs

  • Magma
    [(2^n+1)^2-2 : n in [1..30]]; // Wesley Ivan Hurt, Jul 08 2014
  • Maple
    A093069:=n->(2^n+1)^2-2: seq(A093069(n), n=1..30);
  • Mathematica
    a[ n_] := If[ n < 1, 0, 4^n + 2^(n + 1) - 1]; (* Michael Somos, Jul 08 2014 *)
CoefficientList[Series[(7 - 26*x + 16*x^2)/((1 - x)*(2*x - 1)*(4*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 08 2014 *)
LinearRecurrence[{7,-14,8},{7,23,79},30] (* Harvey P. Dale, Aug 25 2025 *)
  • PARI
    vector(100, n, (2^n+1)^2-2) \\ Colin Barker, Jul 08 2014
  • PARI
    Vec(-(16*x^2-26*x+7)/((x-1)*(2*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Jul 08 2014

Formula

a(n) = 4^n+2^(n+1)-1.
G.f.: -x*(7-26*x+16*x^2) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011
a(n) = A092431(n+2) - A020522(n+1). - R. J. Mathar, Jun 01 2011
E.g.f.: -exp(x) + 2*exp(2*x) + exp(4*x) - 2. - Stefano Spezia, Dec 09 2019

Extensions

More terms from Colin Barker, Jul 08 2014

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 } }.

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

A098879 a(n) = (2^n - 1)^5 - 2.

Original entry on oeis.org

-2, -1, 241, 16805, 759373, 28629149, 992436541, 33038369405, 1078203909373, 34842114263549, 1120413075641341, 35940921946155005, 1151514816750309373, 36870975646169341949, 1180231376725002502141, 37773167607267111108605, 1208833588708967444709373
Offset: 0

Views

Author

Parthasarathy Nambi, Oct 13 2004

Keywords

Comments

5th-power analog of what for exponent 2 is A093112 (2^n-1)^2 - 2 = 4^n - 2^{n+1} - 1 and exponent 3 is A098878 (2^n - 1)^3 - 2. Primes include a(n) for n = 0, 2, 5, 6. These are "near-5th-power prime." Semiprimes include a(n) for n = 3, 8, 9, 10, 13, 15, 21, 29, 33, 40. - Jonathan Vos Post, May 03 2006

Examples

			If n=2, (2^2 - 1)^5 - 2 = 241 (a prime).

Links

Crossrefs

Cf. A091516, A091515, A098878, A091514.

Programs

  • Mathematica
    (2^Range[0,20]-1)^5-2 (* or *) LinearRecurrence[{63,-1302,11160,-41664,64512,-32768},{-2,-1,241,16805,759373,28629149},20] (* Harvey P. Dale, Nov 03 2016 *)
  • PARI
    a(n)=(2^n-1)^5-2 \\ Charles R Greathouse IV, Feb 19 2016

Formula

G.f.: (-2+125*x-2300*x^2+22640*x^3-57728*x^4+66560*x^5)/((-1+x)(-1+32*x)(-1+16*x)(-1+8*x)(-1+4*x)(-1+2*x)). - R. J. Mathar, Nov 14 2007

Extensions

More terms from Jonathan Vos Post, May 03 2006
Edited by N. J. A. Sloane, Sep 30 2007

A117921 Primes of the form (2^k - 1)^3 - 2.

Original entry on oeis.org

3373, 29789, 133432829, 8577357821, 281462092005373
Offset: 1

Views

Author

Jonathan Vos Post, May 03 2006

Keywords

Comments

Exponent-3 analog of what for exponent 2 is A091516 Carol primes (2^n-1)^2 - 2 = 4^n - 2^{n+1} - 1. Hence this is a type of "near-cube primes."

Examples

			a(1) = (2^4 - 1)^3 - 2 = 3373 is prime.
a(2) = (2^5 - 1)^3 - 2 = 29789 is prime.
a(3) = (2^9 - 1)^3 - 2 = 133432829 is prime.
a(4) = (2^11 - 1)^3 - 2 = 8577357821 is prime.

Links

Crossrefs

Cf. A091516, A091515, A098878, A091514.

Programs

  • Mathematica
    Select[(2^Range[20]-1)^3-2,PrimeQ] (* Harvey P. Dale, Oct 22 2016 *)

Formula

A098878 INTERSECTION A000040. {(2^k - 1)^3 - 2 iff prime}.

A118558 a(n) = (2^n-1)^4 - 2.

Original entry on oeis.org

-2, -1, 79, 2399, 50623, 923519, 15752959, 260144639, 4228250623, 68184176639, 1095222947839, 17557851463679, 281200199450623, 4501401006735359, 72040003462430719, 1152780773560811519, 18445618199572250623, 295138898083176775679, 4722294425687923097599
Offset: 0

Views

Author

Jonathan Vos Post, May 03 2006

Keywords

Examples

			a(0) = (2^0 - 1)^4 - 2 = 0^4 - 2 = -2.
a(1) = (2^1 - 1)^4 - 2 = 1^4 - 2 = -1.
a(2) = (2^2 - 1)^4 - 2 = 3^4 - 2 = 79.

Links

Crossrefs

Cf. A091516, A091515, A098878, A091514.
Cf. A093112, A098878.

Programs

Formula

a(n) = (2^n - 1)^4 - 2.
G.f.: (1984*x^4-2120*x^3+510*x^2-61*x+2) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)). - Colin Barker, Apr 30 2013

Extensions

Offset changed to 0 by Paolo Xausa, Apr 19 2024

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
Showing 1-9 of 9 results.

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

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