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

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

Author

Jud McCranie, Dec 09 1999

Keywords

• nonn
• hard

Comments

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]

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A057735, A087885, A014224, A058959.

Programs

Mathematica

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

PARI

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

Extensions

{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