A058959 -id:A058959 - cp's OEIS frontend

A014224 Numbers k such that 3^k - 2 is prime.

2, 4, 5, 6, 9, 22, 37, 41, 90, 102, 105, 317, 520, 541, 561, 648, 780, 786, 957, 1353, 2224, 2521, 6184, 7989, 8890, 19217, 20746, 31722, 37056, 69581, 195430, 225922, 506233, 761457, 1180181

Offset: 1

Jud McCranie

If n is of the form 4k + 3 then 3^n - 2 is composite, because 3^n - 2 = (3^4)^k*3^3 - 2 == 0 (mod 5). So there is no term of the form 4k + 3. If Q is a perfect number such that gcd(3(3^a(n) - 2), Q) = 1 then x = 3^(a(n) - 1)*(3^a(n) - 2)*Q is a solution of the equation sigma(x) = 3x + Q. See comment lines of the sequences A058959 and A171271. -  M. F. Hasler and Farideh Firoozbakht, Dec 07 2009
For all numbers n in this sequence, 3^(n-1)*(3^n-2) is a 2-hyperperfect number, cf. A007593, and no other 2-hyperperfect number seems to be known. - Farideh Firoozbakht and M. F. Hasler, Apr 25 2012
225922 is the last term in the sequence up to 500000. All n <= 500000 have been tested with the Miller-Rabin PRP test and/or PFGW. - Ryan Propper, Aug 18 2013
For n <= 506300 there is one additional term, 506233, a probable prime as tested by PFGW. - Ryan Propper, Sep 03 2013
a(35) > 10^6. - Ryan Propper, Jul 22 2015

Daniel Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem; Vol. 4, No. 2, Dec 1978, pp. 277-302. [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (pp. 114-134) [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing. [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]

Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010. See pp.79-80.
F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Henri and Renaud Lifchitz, PRP Records.

3^n - 2 = A058481(n).

Cf. A058959, A171271. - M. F. Hasler and Farideh Firoozbakht, Dec 07 2009

A014224 = {}; Do[If[PrimeQ[3^n - 2], Print[n]; AppendTo[A014224, n]], {n, 10^5}]; A014224 (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Do[If[Mod[n, 4] != 3 && PrimeQ[3^n - 2], Print[n]], {n, 10000}] (* M. F. Hasler and Farideh Firoozbakht, Dec 07 2009 *)

for(n=1,1e4,if(ispseudoprime(3^n-2),print1(n", "))) \\ Charles R Greathouse IV, Jul 19 2011

Corrected by Andrey V. Kulsha, Feb 04 2001
a(26) = 19217, a(27) = 20746 from Ryan Propper, May 11 2007
a(28) = 31722 from Henri Lifchitz, Oct 2002
a(29) = 37056 from Henri Lifchitz, Oct 2004
a(30) = 69581 from Henri Lifchitz, Jan 2005
a(31) = 195430 from Theodore Burton, Feb 2007
a(32) = 225922 from Ryan Propper, Aug 18 2013
a(33) = 506233 from Ryan Propper, Sep 02 2013
a(34) = 761457 from Ryan Propper, Jul 22 2015
a(35) = 1180181 from Jorge Coveiro, May 22 2020

A051783 Numbers k such that 3^k + 2 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 10, 14, 15, 24, 26, 36, 63, 98, 110, 123, 126, 139, 235, 243, 315, 363, 386, 391, 494, 1131, 1220, 1503, 1858, 4346, 6958, 7203, 10988, 22316, 33508, 43791, 45535, 61840, 95504, 101404, 106143, 107450, 136244, 178428, 361608, 504206, 1753088

Offset: 1

Jud McCranie, Dec 09 1999

From Farideh Firoozbakht and M. F. Hasler, Dec 06 2009: (Start)
If Q is a perfect number such that gcd(Q, 3(3^a(n) + 2)) = 1, then x = 3^(a(n) - 1)*(3^a(n) + 2)*Q is a solution of the equation sigma(x) = 3(x - Q). This is a result of the following theorem:
Theorem: If Q is a (q-1)-perfect number for some prime q, then for all integers t, the equation sigma(x) = q*x - (2t+1)*Q has the solution x = q^(k-1)*p*Q whenever k is a positive integer such that p = q^k + 2t is prime, gcd(q^(k-1), p) = 1 and gcd(q^(k-1)*p,Q) = 1.
Note that by taking t = -1/2(m*q+1), this theorem gives us some solutions of the equation sigma(x) = q *(x + m*Q). See comment lines of the sequence A058959. (End)
No further terms < 200000. - Ray Chandler, Jul 31 2011
A090649 implies that 361608 is a member of this sequence. - Robert Price, Aug 18 2014
No further terms < 320000. - Luke W. Richards, Mar 04 2018
a(45) and a(46) are probable primes because a primality certificate has not yet been found. They have been verified PRP with mprime. - Luke W. Richards, May 04 2018
No further terms < 1300000. - Luke W. Richards, May 17 2018
No further terms < 1400000. - Luke W. Richards, Jul 28 2020
Conjecture: The number n = 3^k + 2 is prime if and only if 2^((n-1)/2) == -1 (mod n). - Maheswara Rao Valluri, Jun 01 2020. [Note that this is an if and only if assertion, so it does not follow from Fermat's Little Theorem. - N. J. A. Sloane, Sep 07 2020]

			3^8 + 2 = 6563 is prime, so 8 is in the sequence.
3^26 + 2 = 2541865828331, a prime number, so 26 is in the sequence.

F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Henri and Renaud Lifchitz, PRP Records.

Cf. A057735, A087885, A014224, A058959.

A051783 = Select[Range[0, 20000], PrimeQ[3^# + 2] &]

is(n)=ispseudoprime(3^n+2) \\ Charles R Greathouse IV, Mar 21 2013

{4346, 6958, 7203} from David J. Rusin, Sep 29 2000
10988 from Ray Chandler, Nov 21 2004
{22316, 33508} found by Henri Lifchitz, Sep-Oct 2002
{43791, 45535, 61840} found by Henri Lifchitz, Oct-Nov 2004
95504 found by Wojciech Florek Dec 15 2005. - Alexander Adamchuk, Mar 02 2008
Edited by N. J. A. Sloane, Dec 19 2009
{101404, 106143, 107450, 136244} from Mike Oakes, Nov 2009
178428 from Ray Chandler, Jul 29 2011
a(45)-a(46) from Luke W. Richards, May 04 2018
a(47) from Paul Bourdelais, Mar 29 2022

A217137 Numbers k such that 3^k + 10 is prime.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 18, 36, 98, 114, 134, 138, 212, 252, 516, 1166, 2321, 2442, 2732, 4569, 8622, 8709, 16487, 22722, 25242, 29928, 32034, 33783, 34001, 44934, 50868, 77861, 90188, 102102, 171843, 178226, 273521

Offset: 1

Vincenzo Librandi, Oct 01 2012

a(37) > 2*10^5. - Robert Price, Oct 23 2013

a(38) > 3*10^5. - Tyler NeSmith, Jan 16 2021

Cf. A014224, A051783, A058958, A058959, A102907.

Select[Range[0, 5000], PrimeQ[3^# + 10] &]

for(n=0, 5*10^3, if(isprime(3^n+10), print1(n", ")))

a(21)-a(36) from Robert Price, Oct 23 2013

a(37) from Tyler NeSmith, Jan 16 2021

A217347 Numbers k such that 3^k - 10 is prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 17, 18, 21, 22, 36, 38, 41, 54, 56, 81, 92, 100, 106, 160, 310, 406, 560, 902, 5549, 9926, 12334, 19374, 19995, 20166, 39609, 62900, 186903, 244461

Offset: 1

Vincenzo Librandi, Oct 01 2012

a(33) > 2*10^5. - Robert Price, Sep 07 2013

a(34) > 3*10^5. - Tyler NeSmith, Oct 03 2022

Cf. A014224, A051783, A058958, A058959.

/* The code produces the sequence up to 560: */ [n: n in [2..800] | IsPrime(3^n - 10)];

Select[Range[2, 5000], PrimeQ[3^# - 10] &]

for(n=2, 5*10^3, if(isprime(3^n-10), print1(n", ")))

a(24)-a(32) from Robert Price, Sep 07 2013

a(33) from Tyler NeSmith, Oct 03 2022

A205647 Numbers k such that 3^k + 16 is prime.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 9, 12, 13, 15, 27, 31, 49, 57, 60, 75, 139, 147, 283, 327, 488, 604, 700, 825, 908, 1051, 1064, 1215, 5319, 9669, 10136, 16675, 25656, 28933, 35864, 47671, 68028, 73380, 186223, 194965, 221649, 233059, 240644, 513007, 543128, 551491, 648872, 989124, 994536

Offset: 1

Jonathan Vos Post, Jan 30 2012

Indices of primes in A205646.

a(50) > 10^6. - Robert Price, Oct 28 2020

			57 is in the sequence because (3^57) + 16 = 1570042899082081611640534579 is prime.

Cf. Sequences of numbers k such that 3^k + m is prime:
  (m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
  (m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
  (m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
  (m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
  (m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
  (m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Select[Range[0, 2000], PrimeQ[3^# + 16] &] (* T. D. Noe, Jan 30 2012 *)

is(n)=isprime(3^n+16) \\ Charles R Greathouse IV, Feb 17 2017

{n: 3^n + 16 is in A000040} = {n: 3^n + 16 is prime} = {n: A000244(n) is prime} = {n: A205646(n) is prime}.

5319 from Nicolas M. Perrault, Nov 10 2012
a(30)-a(40) from Robert Price, Oct 23 2013
a(41) discovered by Lelio R Paula, Nov 2016
a(42)-a(43) from Robert Price, Apr 02 2020
a(44)-a(46) from Robert Price, May 14 2020
a(47)-a(49) from Robert Price, Oct 28 2020

A217136 Numbers k such that 3^k + 8 is prime.

Original entry on oeis.org

1, 2, 4, 5, 8, 13, 14, 20, 38, 44, 77, 88, 124, 152, 244, 557, 2429, 4382, 6268, 18488, 75097, 81998, 96460, 143497

Offset: 1

Vincenzo Librandi, Oct 01 2012

nonn

a(25) > 2*10^5. - Robert Price, Sep 25 2013

Cf. A014224, A051783, A058958, A058959.

Select[Range[0, 5000], PrimeQ[3^# + 8] &]

for(n=1, 5*10^3, if(isprime(3^n+8), print1(n", ")))

a(19)-a(24) from Robert Price, Sep 25 2013

A217135 Numbers k such that 3^k - 8 is prime.

Original entry on oeis.org

3, 4, 7, 8, 14, 20, 22, 62, 139, 254, 272, 430, 907, 1906, 2278, 2827, 3598, 6812, 15266, 20915, 26180, 26342, 27022, 48275, 65186, 69247, 86647

Offset: 1

Vincenzo Librandi, Oct 01 2012

nonn

a(28) > 2*10^5. - Robert Price, Sep 02 2013

Cf. A014224, A051783, A058958, A058959.

Select[Range[2, 5000], PrimeQ[3^# - 8] &]

for(n=2, 5*10^3, if(isprime(3^n-8), print1(n", ")))

a(18)-a(27) from Robert Price, Sep 02 2013

A219035 Numbers k such that 3^k + 14 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 25, 98, 122, 153, 190, 258, 511, 549, 1703, 1790, 1870, 2418, 5226, 5258, 5626, 8550, 13174, 16718, 23669, 25215, 33447, 182566, 188286

Offset: 1

Nicolas M. Perrault, Nov 10 2012

a(31) > 2*10^5. - Robert Price, Sep 27 2013

Cf. Sequences of numbers k such that 3^k + m is prime:
  (m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
  (m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
  (m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
  (m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
  (m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
  (m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Do[If[PrimeQ[3^n + 14], Print[n]], {n, 0, 10000}]

is(n)=isprime(3^n+14) \\ Charles R Greathouse IV, Feb 17 2017

a(24)-a(30) from Robert Price, Sep 27 2013

A219038 Numbers k such that 3^k - 14 is prime.

Original entry on oeis.org

3, 4, 5, 8, 17, 19, 29, 124, 304, 640, 1205, 1549, 1805, 2492, 2945, 13075, 20237, 102763, 173755, 173828, 174040

Offset: 1

Nicolas M. Perrault, Nov 10 2012

a(22) > 2*10^5. - Robert Price, Aug 31 2013

Cf. Sequences of numbers k such that 3^k + m is prime:
  (m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
  (m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
  (m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
  (m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
  (m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
  (m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Do[If[PrimeQ[3^n - 14], Print[n]], {n, 3, 3000}]
Select[Range[1000], PrimeQ[3^# - 14] &] (* Alonso del Arte, Nov 10 2012 *)

is(n)=isprime(3^n-14) \\ Charles R Greathouse IV, Feb 17 2017

a(16)-a(21) from Robert Price, Aug 31 2013

A219040 Numbers k such that 3^k + 20 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 10, 11, 14, 44, 55, 68, 71, 80, 123, 158, 213, 220, 272, 668, 725, 885, 1132, 1677, 2056, 2618, 3130, 3986, 6027, 8660, 11582, 12278, 14054, 62956, 103431, 120434, 123890, 181407

Offset: 1

Nicolas M. Perrault, Nov 10 2012

a(40) > 2*10^5. - Robert Price, Oct 20 2013

			3^3 + 20 = 47 (prime), so 3 is in the sequence.

Cf. Sequences of numbers k such that 3^k + m is prime:
  (m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
  (m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
  (m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
  (m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
  (m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
  (m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Do[If[PrimeQ[3^n + 20], Print[n]], {n, 10000}]

is(n)=isprime(3^n+20) \\ Charles R Greathouse IV, Feb 17 2017

a(32)-a(39) from Robert Price, Oct 20 2013

cp's OEIS Frontend

A014224 Numbers k such that 3^k - 2 is prime.

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A051783 Numbers k such that 3^k + 2 is prime.

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A217137 Numbers k such that 3^k + 10 is prime.

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A217347 Numbers k such that 3^k - 10 is prime.

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A205647 Numbers k such that 3^k + 16 is prime.

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A217136 Numbers k such that 3^k + 8 is prime.

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A217135 Numbers k such that 3^k - 8 is prime.

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A219035 Numbers k such that 3^k + 14 is prime.

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