This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A051783 #100 Apr 03 2022 15:58:28
%S A051783 0,1,2,3,4,8,10,14,15,24,26,36,63,98,110,123,126,139,235,243,315,363,
%T A051783 386,391,494,1131,1220,1503,1858,4346,6958,7203,10988,22316,33508,
%U A051783 43791,45535,61840,95504,101404,106143,107450,136244,178428,361608,504206,1753088
%N A051783 Numbers k such that 3^k + 2 is prime.
%C A051783 From _Farideh Firoozbakht_ and _M. F. Hasler_, Dec 06 2009: (Start)
%C A051783 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 A051783 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 A051783 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 A051783 No further terms < 200000. - _Ray Chandler_, Jul 31 2011
%C A051783 A090649 implies that 361608 is a member of this sequence. - _Robert Price_, Aug 18 2014
%C A051783 No further terms < 320000. - _Luke W. Richards_, Mar 04 2018
%C A051783 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 A051783 No further terms < 1300000. - _Luke W. Richards_, May 17 2018
%C A051783 No further terms < 1400000. - _Luke W. Richards_, Jul 28 2020
%C A051783 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]
%H A051783 F. Firoozbakht and 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 A051783 Henri and Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=3%5En%2B2&action=Search">PRP Records</a>.
%e A051783 3^8 + 2 = 6563 is prime, so 8 is in the sequence.
%e A051783 3^26 + 2 = 2541865828331, a prime number, so 26 is in the sequence.
%t A051783 A051783 = Select[Range[0, 20000], PrimeQ[3^# + 2] &]
%o A051783 (PARI) is(n)=ispseudoprime(3^n+2) \\ _Charles R Greathouse IV_, Mar 21 2013
%Y A051783 Cf. A057735, A087885, A014224, A058959.
%K A051783 nonn,hard
%O A051783 1,3
%A A051783 _Jud McCranie_, Dec 09 1999
%E A051783 {4346, 6958, 7203} from _David J. Rusin_, Sep 29 2000
%E A051783 10988 from _Ray Chandler_, Nov 21 2004
%E A051783 {22316, 33508} found by _Henri Lifchitz_, Sep-Oct 2002
%E A051783 {43791, 45535, 61840} found by _Henri Lifchitz_, Oct-Nov 2004
%E A051783 95504 found by Wojciech Florek Dec 15 2005. - _Alexander Adamchuk_, Mar 02 2008
%E A051783 Edited by _N. J. A. Sloane_, Dec 19 2009
%E A051783 {101404, 106143, 107450, 136244} from _Mike Oakes_, Nov 2009
%E A051783 178428 from _Ray Chandler_, Jul 29 2011
%E A051783 a(45)-a(46) from _Luke W. Richards_, May 04 2018
%E A051783 a(47) from _Paul Bourdelais_, Mar 29 2022