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Search: a093069 -id:a093069
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A091513 Numbers k such that (2^k + 1)^2 - 2 = 4^k + 2^(k+1) - 1 is prime. +10
10
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..51.
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
FORMULA
A093069(n) = (2^a(n) + 1)^2 - 2.
MATHEMATICA
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 *)
PROG
(Magma) [n: n in [0..500] | IsPrime((2^n+1)^2-2)]; // Vincenzo Librandi, Feb 19 2016
(PARI) is(n)=ispseudoprime((2^n+1)^2-2) \\ Charles R Greathouse IV, Feb 19 2016
CROSSREFS
Cf. A091514 (primes of the form (2^n + 1)^2 - 2).
Cf. A093069 (numbers of the form (2^n + 1)^2 - 2).
KEYWORD
nonn,hard
AUTHOR
Eric W. Weisstein, Jan 17 2004
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)
STATUS
approved
A091514 Primes of the form (2^n + 1)^2 - 2 = 4^n + 2^(n+1) - 1. +10
10
2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207, 5070602400912922109586440191999
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Cletus Emmanuel calls these "Kynea primes".
LINKS
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
FORMULA
a(n) = (2^A091513(n) + 1)^2 - 2.
MAPLE
select(isprime, [seq((2^n+1)^2-2, n=0..1000)]); # Robert Israel, Feb 10 2016
MATHEMATICA
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 *)
PROG
(Magma) [a: n in [0..60] | IsPrime(a) where a is 4^n+2^(n+1)-1]; // Vincenzo Librandi, Dec 13 2011
(PARI) select(isprime, vector(100, n, (2^n+1)^2-2)) \\ Charles R Greathouse IV, Feb 19 2016
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).
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jan 17 2004
EXTENSIONS
Edited by Ray Chandler, Nov 15 2004
First term (2) added by Vincenzo Librandi, Dec 13 2011
STATUS
approved
A093112 a(n) = (2^n-1)^2 - 2. +10
7
-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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Cletus Emmanuel calls these "Carol numbers".
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1660
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).
FORMULA
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
MAPLE
seq((Stirling2(n+1, 2))^2-2, n=1..23); # Zerinvary Lajos, Dec 20 2006
MATHEMATICA
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 *)
PROG
(PARI) Vec(x*(16*x^2-14*x+1)/((x-1)*(2*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Jul 07 2014
(PARI) a(n) = (2^n-1)^2-2 \\ Charles R Greathouse IV, Sep 10 2015
(Python)
def A093112(n): return (2**n-1)**2-2 # Chai Wah Wu, Feb 18 2022
CROSSREFS
Cf. A000225.
KEYWORD
sign,easy
AUTHOR
Eric W. Weisstein, Mar 20 2004
EXTENSIONS
More terms from Colin Barker, Jul 07 2014
STATUS
approved
A360994 Numbers k such that (2^k + 1)^3 - 2 is a semiprime. +10
3
0, 1, 2, 4, 5, 6, 13, 14, 18, 27, 43, 45, 63, 76, 85, 108, 115, 119, 123, 187, 211, 215, 283, 312
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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
Table of n, a(n) for n=1..24.
factordb.com, Status of (2^355+1)^3-2.
FORMULA
{ k >= 0 : A099359(k) in { A001358 } }.
MATHEMATICA
Select[Range[70], PrimeOmega[(2^# + 1)^3 - 2] == 2 &]
PROG
(Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [n: n in [1..70]| IsSemiprime(s) where s is (2^n+1)^3-2];
(PARI) isok(n) = bigomega((2^n+1)^3-2) == 2;
CROSSREFS
Cf. A001358, A091513, A091514, A093069, A099359, A100899, A100900, A269264, A360993.
KEYWORD
nonn,more,hard
AUTHOR
Serge Batalov, Feb 27 2023
STATUS
approved
A100496 Numbers n such that (2^n+1)^4-2 is prime. +10
2
1, 7, 25, 31, 34, 271, 514, 2896, 8827, 16816, 37933
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
LINKS
Table of n, a(n) for n=1..11.
EXAMPLE
a(1) = 1 because (2^1+1)^4 - 2 = 79 is prime and is the first such prime.
MATHEMATICA
Select[Range[5000], PrimeQ[(2^# + 1)^4 - 2] &]
PROG
(PARI) is(n)=ispseudoprime((2^n+1)^4-2) \\ Charles R Greathouse IV, Jun 13 2017
CROSSREFS
Cf. A091513, A091514, A093069, A099359.
Cf. A100497, n such that (2^n+1)^4-2 is semiprime.
KEYWORD
more,nonn
AUTHOR
Jonathan Vos Post, Nov 23 2004
EXTENSIONS
Edited, corrected and extended by Ray Chandler and Hugo Pfoertner, Nov 26 2004
a(10)-a(11) from Tyler Busby, Feb 12 2023
STATUS
approved
A244663 Binary representation of 4^n + 2^(n+1) - 1. +10
2
111, 10111, 1001111, 100011111, 10000111111, 1000001111111, 100000011111111, 10000000111111111, 1000000001111111111, 100000000011111111111, 10000000000111111111111, 1000000000001111111111111, 100000000000011111111111111, 10000000000000111111111111111
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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).
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)).
EXAMPLE
a(3) is 1001111 because A093069(3) = 79 which is 1001111 in base 2.
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 *)
PROG
(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))
(Magma) [-1/9 + 10^(1 + n)/9 + 100^n : n in [1..15]]; // Wesley Ivan Hurt, Jul 09 2014
CROSSREFS
Cf. A093069.
KEYWORD
nonn,easy,base
AUTHOR
Colin Barker, Jul 08 2014
STATUS
approved
A268574 Numbers k such that (2^k + 1)^2 - 2 is a semiprime. +10
2
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(48) >= 428. - Serge Batalov, Feb 25 2023
LINKS
Table of n, a(n) for n=1..47.
factordb.com, Status of (2^428+1)^2-2.
EXAMPLE
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.
MATHEMATICA
Select[Range[105], PrimeOmega[(2^# + 1)^2 - 2] == 2 &]
PROG
(Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [n: n in [1..110]| IsSemiprime(s) where s is (2^n+1)^2-2];
(PARI) isok(n) = bigomega((2^n+1)^2-2) == 2; \\ Michel Marcus, Feb 22 2016
CROSSREFS
Cf. A091513, A091514, A093069, A269264, A360993, A360994.
KEYWORD
nonn,more,hard
AUTHOR
Vincenzo Librandi, Feb 21 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
STATUS
approved
A130567 Expansion of x*(2 - 7*x + 2*x^2)/((1-x)*(1-4*x)*(1-2*x)). +10
0
2, 7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247, 268468223, 1073807359, 4295098367, 17180131327, 68720001023, 274878955519, 1099513724927, 4398050705407, 17592194433023, 70368760954879
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..24.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).
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)
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 *)
CROSSREFS
Cf. A093069, A099393, A028244.
KEYWORD
nonn,easy,changed
AUTHOR
Roger L. Bagula, Aug 09 2007
EXTENSIONS
New name from Joerg Arndt, Feb 08 2015
STATUS
approved
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Last modified September 19 17:13 EDT 2024. Contains 376014 sequences. (Running on oeis4.)