A093112 -id:A093112 - cp's OEIS frontend

A091515 Numbers k such that (2^k - 1)^2 - 2 = 4^k - 2^(k+1) - 1 is prime.

2, 3, 4, 6, 7, 10, 12, 15, 18, 19, 21, 25, 27, 55, 129, 132, 159, 171, 175, 315, 324, 358, 393, 435, 786, 1459, 1707, 2923, 6462, 14289, 39012, 51637, 100224, 108127, 110953, 175749, 185580, 226749, 248949, 253987, 520363, 653490, 688042, 695631

Offset: 1

Eric W. Weisstein, Jan 17 2004

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

Cf. A093112, A091516.

lst={};Do[p=(2^n-1)^2-2;If[PrimeQ[p],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)

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

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004
a(36)=175749 from Cletus Emmanuel (cemmanu(AT)yahoo.com), Oct 08 2004
a(37)=185580 from Cletus Emmanuel (cemmanu(AT)yahoo.com), Nov 03 2004
Edited by Ray Chandler, Nov 15 2004
a(38)=226749 from Steven Harvey, Jan 11 2005 and subsequently confirmed as next term
a(39) from Eric W. Weisstein, Mar 31 2006
a(40) = 253987 from Cletus Emmanuel (cemmanu(AT)yahoo.com), May 03 2007
a(41) = 520363 from Eric W. Weisstein, Jun 08 2016 (computed by Mark Rodenkirch)
a(42) = 653490 from Eric W. Weisstein, Jun 15 2016 (computed by Mark Rodenkirch)
a(43) = 688042 from Mark Rodenkirch, Jul 05 2016
a(44) = 695631 from Mark Rodenkirch, Jul 16 2016

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

Eric W. Weisstein, Mar 17 2004

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

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

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).

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

Cf. A244663.

[(2^n+1)^2-2 : n in [1..30]]; // Wesley Ivan Hurt, Jul 08 2014

A093069:=n->(2^n+1)^2-2: seq(A093069(n), n=1..30);

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 *)

vector(100, n, (2^n+1)^2-2) \\ Colin Barker, Jul 08 2014

Vec(-(16*x^2-26*x+7)/((x-1)*(2*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Jul 08 2014

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

More terms from Colin Barker, Jul 08 2014

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

Parthasarathy Nambi, Oct 13 2004

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

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

Harvey P. Dale, Table of n, a(n) for n = 0..664
Eric Weisstein's World of Mathematics, Near-Square Prime.
Index entries for linear recurrences with constant coefficients, signature (63, -1302, 11160, -41664, 64512, -32768).

Cf. A091516, A091515, A098878, A091514.

(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 *)

a(n)=(2^n-1)^5-2 \\ Charles R Greathouse IV, Feb 19 2016

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

More terms from Jonathan Vos Post, May 03 2006

Edited by N. J. A. Sloane, Sep 30 2007

A244845 Binary representation of 4^n - 2^(n+1) - 1.

Original entry on oeis.org

111, 101111, 11011111, 1110111111, 111101111111, 11111011111111, 1111110111111111, 111111101111111111, 11111111011111111111, 1111111110111111111111, 111111111101111111111111, 11111111111011111111111111, 1111111111110111111111111111

Offset: 2

Colin Barker, Jul 07 2014

			a(3) is 101111 because A093112(3) = 47 which is 101111 in base 2.

Colin Barker, Table of n, a(n) for n = 2..500
Wikipedia, Carol number
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A093112.

Table[FromDigits[IntegerDigits[4^n-2^(n+1)-1,2]],{n,2,15}] (* Harvey P. Dale, Oct 03 2016 *)

vector(100, n, (100^(n+1)-9*10^(2+n)-1)/9)

Vec(x^2*(89000*x^2-88790*x-111)/((x-1)*(10*x-1)*(100*x-1)) + O(x^100))

a(n) = subst(Pol(binary(4^n-2^(n+1)-1)), x, 10); \\ Michel Marcus, Jul 08 2014

a(n) = 111*a(n-1)-1110*a(n-2)+1000*a(n-3).
a(n) = (-1-9*10^(1+n)+100^n)/9.
G.f.: x^2*(89000*x^2-88790*x-111) / ((x-1)*(10*x-1)*(100*x-1)).

A331858 a(n) = (2^p-1)(2^(p-1))((2^p-1)^2-2), where p is the n-th prime.

Original entry on oeis.org

42, 1316, 475664, 131080256, 8783210218496, 2250975213522944, 147570574898545885184, 37778715690312487141376, 2475879193127080196116054016, 41538374636164863806350357434466304, 10633823951424046514111736193740701696, 178405961584350762488394070192754827810832384

Offset: 1

G. L. Honaker, Jr., Jan 29 2020

Integers a(1), a(2), a(4), a(8) corresponding to p = 2, 3, 7, 19 are also terms of A331805. - Bernard Schott, Feb 04 2020

Cf. A000040 (primes), A000396 (perfect numbers), A093112 ((2^n-1)^2-2), A060286 (2^(p-1)*(2^p-1)), A331805.

f[p_] := (2^p-1)*(2^(p-1))*((2^p-1)^2-2); f @ Prime @ Range[12] (* Amiram Eldar, Jan 29 2020 *)

[(2^p-1)*((2^p-1)^2-2)<<(p-1) | p<-primes(12)] \\ or: a(n,p=prime(n))={...}. - M. F. Hasler, Jan 29 2020

a(n) = A060286(n)*A093112(prime(n)). - M. F. Hasler, Jan 31 2020

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

Jonathan Vos Post, May 03 2006

			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.

Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024).

Cf. A091516, A091515, A098878, A091514.

Cf. A093112, A098878.

(2^Range[0, 20] - 1)^4 - 2 (* Paolo Xausa, Apr 19 2024 *)

a(n)=(2^n-1)^4-2 \\ Charles R Greathouse IV, Feb 19 2016

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

Offset changed to 0 by Paolo Xausa, Apr 19 2024

A381698 Numbers of the form (2^k-1)^2 - 2 that are squarefree.

Original entry on oeis.org

7, 47, 223, 959, 3967, 16127, 261119, 1046527, 4190207, 16769023, 67092479, 268402687, 1073676287, 4294836223, 17179607039, 68718952447, 274876858367, 1099509530623, 4398042316799, 17592177655807, 70368727400447, 281474943156223, 1125899839733759, 4503599493152767, 18014398241046527

Offset: 1

Massimo Kofler, Mar 04 2025

nonn

Intersection of A093112 and A005117.

Cf. A000225.

Select[Table[(2^n-1)^2 - 2, {n, 2, 30}], SquareFreeQ] (* Amiram Eldar, Mar 04 2025 *)

select(issquarefree, vector(30, n, n++; (2^n-1)^2 - 2)) \\ Michel Marcus, Mar 04 2025

cp's OEIS Frontend

A091515 Numbers k such that (2^k - 1)^2 - 2 = 4^k - 2^(k+1) - 1 is prime.

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

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

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A244845 Binary representation of 4^n - 2^(n+1) - 1.

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A331858 a(n) = (2^p-1)*(2^(p-1))*((2^p-1)^2-2), where p is the n-th prime.

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

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A381698 Numbers of the form (2^k-1)^2 - 2 that are squarefree.

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A331858 a(n) = (2^p-1)(2^(p-1))((2^p-1)^2-2), where p is the n-th prime.