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%I #76 Nov 09 2023 07:39:41
%S 2,4,5,6,9,22,37,41,90,102,105,317,520,541,561,648,780,786,957,1353,
%T 2224,2521,6184,7989,8890,19217,20746,31722,37056,69581,195430,225922,
%U 506233,761457,1180181
%N Numbers k such that 3^k - 2 is prime.
%C 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
%C 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
%C 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
%C For n <= 506300 there is one additional term, 506233, a probable prime as tested by PFGW. - _Ryan Propper_, Sep 03 2013
%C a(35) > 10^6. - _Ryan Propper_, Jul 22 2015
%D 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]
%D 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]
%D 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]
%H Antal Bege and Kinga Fogarasi, <a href="https://arxiv.org/abs/1008.0155">Generalized perfect numbers</a>, arXiv:1008.0155 [math.NT], 2010. See pp.79-80.
%H 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 Henri and Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=3%5En-2&action=Search">PRP Records</a>.
%t A014224 = {}; Do[If[PrimeQ[3^n - 2], Print[n]; AppendTo[A014224, n]], {n, 10^5}]; A014224 (* _Vladimir Joseph Stephan Orlovsky_, Aug 21 2008 *)
%t Do[If[Mod[n, 4] != 3 && PrimeQ[3^n - 2], Print[n]], {n, 10000}] (* _M. F. Hasler_ and _Farideh Firoozbakht_, Dec 07 2009]
%o (PARI) for(n=1,1e4,if(ispseudoprime(3^n-2),print1(n", "))) \\ _Charles R Greathouse IV_, Jul 19 2011
%Y 3^n - 2 = A058481(n).
%Y Cf. A058959, A171271. - _M. F. Hasler_ and _Farideh Firoozbakht_, Dec 07 2009
%K nonn,more
%O 1,1
%A _Jud McCranie_
%E Corrected by _Andrey V. Kulsha_, Feb 04 2001
%E a(26) = 19217, a(27) = 20746 from _Ryan Propper_, May 11 2007
%E a(28) = 31722 from _Henri Lifchitz_, Oct 2002
%E a(29) = 37056 from _Henri Lifchitz_, Oct 2004
%E a(30) = 69581 from _Henri Lifchitz_, Jan 2005
%E a(31) = 195430 from Theodore Burton, Feb 2007
%E a(32) = 225922 from _Ryan Propper_, Aug 18 2013
%E a(33) = 506233 from _Ryan Propper_, Sep 02 2013
%E a(34) = 761457 from _Ryan Propper_, Jul 22 2015
%E a(35) = 1180181 from _Jorge Coveiro_, May 22 2020
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