A091516 -id:A091516 - 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

A269264 Numbers k such that (2^k-1)^2 - 2 is a semiprime.

Original entry on oeis.org

5, 9, 13, 16, 23, 24, 28, 35, 37, 38, 40, 47, 48, 51, 52, 57, 61, 65, 67, 70, 79, 83, 84, 85, 88, 90, 102, 111, 144, 148, 157, 162, 168, 169, 177, 181, 190, 237, 246, 298, 308, 346

Offset: 1

Vincenzo Librandi, Feb 21 2016

a(43) >= 385. - Serge Batalov, Feb 26 2023

			a(1) = 5 because 31^2-2 = 959 = 7*137 which is semiprime.
a(2) = 9 because 511^2-2 = 261119 = 23*11353 which is semiprime.

factordb.com, Status of (2^385-1)^2-2.

Cf. A091515, A091516, A268574, A360993, A360994.

IsSemiprime:=func; [n: n in [2..120]| IsSemiprime(s) where s is (2^n-1)^2-2];

Mathematica

Select[Range[120], PrimeOmega[(2^# - 1)^2 - 2] == 2 &]

PARI

isok(n) = bigomega((2^n-1)^2-2) == 2; \\ Michel Marcus, Feb 22 2016

Extensions

a(29)-a(42) from Hugo Pfoertner, Aug 05 2019

A360993 Numbers k such that (2^k - 1)^3 + 2 is a semiprime.

Original entry on oeis.org

4, 5, 8, 12, 13, 18, 20, 29, 38, 56, 60, 62, 76, 82, 101, 118, 202, 210, 230, 276, 328, 332, 336, 338, 368

Offset: 1

Serge Batalov, Feb 27 2023

a(26) >= 406.
438, 500, 526, 604, 648, 696 are also in this sequence, but their positions cannot be established before finding any factor for the values corresponding to the following "blockers": 406, 496, 528.
2382, 2733, 2910, 3368, 3508, 5338, 7705, 11185, 19905, 23814, 38545, 179294 are larger terms of this sequence, but their positions cannot be established. These produce "trivial" semiprimes where one prime is small (e.g., 3 or 11).

			a(1) = 4 because 15^3 + 2 = 3377 = 11 * 307, which is semiprime.
a(2) = 5 because 31^3 + 2 = 29793 = 3 * 9931, which is semiprime.

factordb.com, Status of (2^406-1)^3+2.

Cf. A091515, A091516, A100899, A100900, A268574, A269264, A360994.

IsSemiprime:=func; [n: n in [2..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;

Extensions

a(20)-a(26) from Serge Batalov, Mar 03 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

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

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

Original entry on oeis.org

3373, 29789, 133432829, 8577357821, 281462092005373

Offset: 1

Jonathan Vos Post, May 03 2006

nonn

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."

			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.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A091516, A091515, A098878, A091514.

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

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

A331805 Integers k such that k is equal to the sum of the nonprime proper divisors of k.

Original entry on oeis.org

42, 1316, 131080256

Offset: 1

G. L. Honaker, Jr., Jan 26 2020

The number 37778715690312487141376 is also in the sequence. - Daniel Suteu, Jan 27 2020
The first 3 terms have the form (2^p-1)*(2^(p-1))*((2^p-1)^2-2), i.e., a Perfect number times a Carol prime. - G. L. Honaker, Jr., Jan 27 2020
In other words, the values of p are given by the intersection of A091515 and A000043. Currently, only four such values of p are known: {2, 3, 7, 19}. - Daniel Suteu, Jan 27 2020
From Bernard Schott, Jan 28 2020: (Start)
Proposition: If a number N_p is of the form Q_p * C_p where Q_p = (2^(p-1)) * (2^p - 1) is a perfect number and C_p = (2^p-1)^2-2 is a Carol prime then, the sum of the nonprime proper divisors of N_p called S_p(N_p) is equal to N_p.
Proof:
The sum of the nonprime proper divisors of N_p is:
S_p(N_p) = (2* Q_p - 2 - (2^p-1)) + ((Q_p - 1) * C_p).
In the first parenthesis, there is the sum of the nonprime proper divisors of N_p coming only from the perfect number Q_p, then in the second parenthesis, there is the sum of the nonprime proper divisors of N_p coming from C_p.
Then, this sum of the nonprime proper divisors of N_p, S_p(N_p) is indeed equal to N_p = (2^(p-1)) * (2^p-1) * ((2^p-1)^2-2).
Hence, (2^19-1)*(2^(19-1))*((2^19-1)^2-2) = 37778715690312487141376 is a term. (End)
10^13 < a(4) <= 72872313094554244192 = 2^5 * 109 * 151 * 65837 * 2101546957. - Giovanni Resta, Jan 28 2020

			42 is a term because 42 = 1 + 6 + 14 + 21.
1316 is a term because 1316 = 1 + 4 + 14 + 28 + 94 + 188 + 329 + 658.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 42
Wikipedia, 42 (number)

Cf. A001065, A018252, A023890, A331858.

Cf. A000043, A091515, A091516 (Carol primes).

fun[p_, e_] := (p^(e+1) - 1)/(p - 1); npsigma[n_] := Times @@ fun @@@ (f = FactorInteger[n]) - Plus @@ First /@ f;; Select[Range[2, 1500], npsigma[#] == 2# &] (* Amiram Eldar, Jan 26 2020 *)

isok(n) = sigma(n) - n - vecsum(factor(n)[,1]) == n; \\ Daniel Suteu, Jan 27 2020

a(2) from Chuck Gaydos

a(3) from Amiram Eldar, Jan 26 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

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

Juri-Stepan Gerasimov, Mar 01 2010

nonn

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

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

Cf. A091513, A091515, A091514, A091516.

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

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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A269264 Numbers k such that (2^k-1)^2 - 2 is a semiprime.

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A360993 Numbers k such that (2^k - 1)^3 + 2 is a semiprime.

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Magma

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

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

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A331805 Integers k such that k is equal to the sum of the nonprime proper divisors of k.

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Mathematica

PARI

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

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