

A051783


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


46



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
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OFFSET

1,3


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 (q1)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^(k1)*p*Q whenever k is a positive integer such that p = q^k + 2t is prime, gcd(q^(k1), p) = 1 and gcd(q^(k1)*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^((n1)/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]


LINKS

Table of n, a(n) for n=1..46.
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Henri & Renaud Lifchitz, PRP Records.


EXAMPLE

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.


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A057735, A087885, A014224, A058959.
Sequence in context: A135772 A209064 A222264 * A033083 A302406 A328092
Adjacent sequences: A051780 A051781 A051782 * A051784 A051785 A051786


KEYWORD

nonn,hard


AUTHOR

Jud McCranie, Dec 09 1999


EXTENSIONS

{4346, 6958, 7203} from David J. Rusin, Sep 29 2000
10988 from Ray Chandler, Nov 21 2004
{22316, 33508} found by Henri Lifchitz, SepOct 2002
{43791, 45535, 61840} found by Henri Lifchitz, OctNov 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


STATUS

approved



