A093069 -id:A093069 - cp's OEIS frontend

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

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

Eric W. Weisstein, Jan 17 2004

S. Harvey, Carol and Kynea Primes
M. Rodenkirch, Carol and Kynea Prime Search
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Near-Square Prime

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

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

[n: n in [0..500] | IsPrime((2^n+1)^2-2)]; // Vincenzo Librandi, Feb 19 2016

Flatten[Position[Table[(2^n + 1)^2 - 2, {n, 0, 10^3}], ?PrimeQ] - 1] (* _Eric W. Weisstein, Feb 10 2016 *)
Select[Range[0, 5000], PrimeQ[(2^# + 1)^2 - 2] & ] (* Vincenzo Librandi, Feb 19 2016 *)

is(n)=ispseudoprime((2^n+1)^2-2) \\ Charles R Greathouse IV, Feb 19 2016

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

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)

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

Eric W. Weisstein, Jan 17 2004

nonn

Cletus Emmanuel calls these "Kynea primes".

Vincenzo Librandi, Table of n, a(n) for n = 1..30
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Near-Square Prime

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

Cf. A091513 (indices n such that (2^n + 1)^2 - 2 is prime).

[a: n in [0..60] | IsPrime(a) where a is 4^n+2^(n+1)-1]; // Vincenzo Librandi, Dec 13 2011

select(isprime,[seq((2^n+1)^2-2, n=0..1000)]); # Robert Israel, Feb 10 2016

lst={};Do[If[PrimeQ[p=4^n+2^(n+1)-1], (*Print[p];*)AppendTo[lst, p]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Table[(2^n + 1)^2 - 2, {n, 0, 50}], PrimeQ] (* Eric W. Weisstein, Feb 10 2016 *)

select(isprime, vector(100,n,(2^n+1)^2-2)) \\ Charles R Greathouse IV, Feb 19 2016

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

Edited by Ray Chandler, Nov 15 2004

First term (2) added by Vincenzo Librandi, Dec 13 2011

A093112 a(n) = (2^n-1)^2 - 2.

Original entry on oeis.org

-1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, 4190207, 16769023, 67092479, 268402687, 1073676287, 4294836223, 17179607039, 68718952447, 274876858367, 1099509530623, 4398042316799, 17592177655807, 70368727400447, 281474943156223, 1125899839733759

Offset: 1

Eric W. Weisstein, Mar 20 2004

Cletus Emmanuel calls these "Carol numbers".

Michael De Vlieger, Table of n, a(n) for n = 1..1660
P. Shanmuganandham and C. Deepa, Sum of Squares of n Consecutive Carol Numbers, Baghdad Science Journal, Vol. 20, No. 1 (Special Issue: ICAAM), 2023, pp. 263-267.
Amelia Carolina Sparavigna, Binary Operators of the Groupoids of OEIS A093112 and A093069 Numbers(Carol and Kynea Numbers), Department of Applied Science and Technology, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Near-Square Prime
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A000225.

seq((Stirling2(n+1, 2))^2-2, n=1..23); # Zerinvary Lajos, Dec 20 2006

lst={};Do[p=(2^n-1)^2-2;AppendTo[lst, p], {n, 66}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
Rest@ CoefficientList[Series[x (16 x^2 - 14 x + 1)/((x - 1) (2 x - 1) (4 x - 1)), {x, 0, 25}], x] (* Michael De Vlieger, Dec 09 2019 *)

Vec(x*(16*x^2-14*x+1)/((x-1)*(2*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Jul 07 2014

a(n) = (2^n-1)^2-2 \\ Charles R Greathouse IV, Sep 10 2015

def A093112(n): return (2**n-1)**2-2 # Chai Wah Wu, Feb 18 2022

a(n) = (2^n-1)^2 - 2.
From Colin Barker, Jul 07 2014: (Start)
a(n) = 6*a(n-1) - 7*a(n-2) - 6*a(n-3) + 8*a(n-4).
G.f.: x*(16*x^2-14*x+1) / ((x-1)*(2*x-1)*(4*x-1)). (End)
E.g.f.: 2 - exp(x) - 2*exp(2*x) + exp(4*x). - Stefano Spezia, Dec 09 2019

More terms from Colin Barker, Jul 07 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

Serge Batalov, Feb 27 2023

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).

factordb.com, Status of (2^355+1)^3-2.

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

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

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Author

Jonathan Vos Post, Nov 23 2004

Keywords

more
nonn

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

A244663 Binary representation of 4^n + 2^(n+1) - 1.

Original entry on oeis.org

111, 10111, 1001111, 100011111, 10000111111, 1000001111111, 100000011111111, 10000000111111111, 1000000001111111111, 100000000011111111111, 10000000000111111111111, 1000000000001111111111111, 100000000000011111111111111, 10000000000000111111111111111

Offset: 1

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Colin Barker, Jul 08 2014

Keywords

nonn
easy
base

Examples

a(3) is 1001111 because A093069(3) = 79 which is 1001111 in base 2.

Links

Colin Barker, Table of n, a(n) for n = 1..450

Wikipedia, Kynea number

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A093069.

Programs

Magma

[-1/9 + 10^(1 + n)/9 + 100^n : n in [1..15]]; // Wesley Ivan Hurt, Jul 09 2014

Maple

A244663:=n->-1/9+10^(1+n)/9+100^n: seq(A244663(n), n=1..15); # Wesley Ivan Hurt, Jul 09 2014

Mathematica

Table[-1/9 + 10^(1 + n)/9 + 100^n, {n, 15}] (* Wesley Ivan Hurt, Jul 09 2014 *) LinearRecurrence[{111,-1110,1000},{111,10111,1001111},20] (* Harvey P. Dale, Dec 11 2014 *)

PARI

vector(100, n, -1/9+10^(1+n)/9+100^n)

PARI

Vec(-x*(2000*x^2-2210*x+111)/((x-1)*(10*x-1)*(100*x-1)) + O(x^100))

Formula

a(n) = -1/9+10^(1+n)/9+100^n.

a(n) = 111*a(n-1)-1110*a(n-2)+1000*a(n-3).

G.f.: -x*(2000*x^2-2210*x+111) / ((x-1)*(10*x-1)*(100*x-1)).

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

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Author

Vincenzo Librandi, Feb 21 2016

Keywords

nonn
more
hard

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

factordb.com, Status of (2^428+1)^2-2.

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

A130567 Expansion of x*(2 - 7*x + 2*x^2)/((1-x)*(1-4*x)*(1-2*x)).

Original entry on oeis.org

2, 7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247, 268468223, 1073807359, 4295098367, 17180131327, 68720001023, 274878955519, 1099513724927, 4398050705407, 17592194433023, 70368760954879

Offset: 1

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Author

Roger L. Bagula, Aug 09 2007

Keywords

nonn
easy

Links

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Cf. A093069, A099393, A028244.

Programs

Mathematica

f[n_Integer?Positive] := f[n] = 2^(2*n - 1) + 2*f[n - 1] + 1; f[0] = 2; Table[f[n], {n, 0, 30}] CoefficientList[Series[x*(2-7x+2x^2)/((1-x)(1-4x)(1-2x)),{x,0,30}],x] (* Harvey P. Dale, Sep 07 2015 *)

Formula

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

From R. J. Mathar, Jun 13 2008: (Start)

O.g.f.: x*(2 - 7*x + 2*x^2)/((1-x)*(1-4*x)*(1-2*x)).

a(n) = A093069(n-2), n>1. (End)

Extensions

New name from Joerg Arndt, Feb 08 2015

Showing 1-8 of 8 results.

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cp's OEIS Frontend

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

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Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A093112 a(n) = (2^n-1)^2 - 2.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A244663 Binary representation of 4^n + 2^(n+1) - 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

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

A130567 Expansion of x(2 - 7x + 2x^2)/((1-x)(1-4x)(1-2*x)).