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 A051783 Numbers k such that 3^k + 2 is prime. 44

%I

%S 0,1,2,3,4,8,10,14,15,24,26,36,63,98,110,123,126,139,235,243,315,363,

%T 386,391,494,1131,1220,1503,1858,4346,6958,7203,10988,22316,33508,

%U 43791,45535,61840,95504,101404,106143,107450,136244,178428,361608,504206

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

%C From _Farideh Firoozbakht_ and _M. F. Hasler_, Dec 06 2009: (Start)

%C 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:

%C 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.

%C 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)

%C No further terms < 200000. - _Ray Chandler_, Jul 31 2011

%C A090649 implies that 361608 is a member of this sequence. - _Robert Price_, Aug 18 2014

%C No further terms < 320000. - _Luke W. Richards_, Mar 04 2018

%C 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

%C No further terms < 1300000. - _Luke W. Richards_, May 17 2018

%H F. Firoozbakht, M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1

%H Henri & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=3%5En%2B2&amp;action=Search">PRP Records</a>.

%e 3^8 + 2 = 6563 is prime, so 8 is in the sequence.

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

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

%o (PARI) is(n)=ispseudoprime(3^n+2) \\ _Charles R Greathouse IV_, Mar 21 2013

%Y Cf. A057735, A087885, A014224, A058959.

%K nonn,hard

%O 1,3

%A _Jud McCranie_, Dec 09 1999

%E {4346, 6958, 7203} from _David J. Rusin_, Sep 29 2000

%E 10988 from _Ray Chandler_, Nov 21 2004

%E {22316, 33508} found by _Henri Lifchitz_, Sep-Oct 2002

%E {43791, 45535, 61840} found by _Henri Lifchitz_, Oct-Nov 2004

%E 95504 found by Wojciech Florek Dec 15 2005. - _Alexander Adamchuk_, Mar 02 2008

%E Edited by _N. J. A. Sloane_, Dec 19 2009

%E {101404, 106143, 107450, 136244} from _Mike Oakes_, Nov 2009

%E 178428 from _Ray Chandler_, Jul 29 2011

%E a(45)-a(46) from _Luke W. Richards_, May 04 2018

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Last modified July 19 21:21 EDT 2018. Contains 312793 sequences. (Running on oeis4.)