A014224 Numbers k such that 3^k - 2 is prime.

2, 4, 5, 6, 9, 22, 37, 41, 90, 102, 105, 317, 520, 541, 561, 648, 780, 786, 957, 1353, 2224, 2521, 6184, 7989, 8890, 19217, 20746, 31722, 37056, 69581, 195430, 225922, 506233, 761457, 1180181

Offset: 1

Author

Jud McCranie

Keywords

• nonn
• more

Comments

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

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

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

For n <= 506300 there is one additional term, 506233, a probable prime as tested by PFGW. - Ryan Propper, Sep 03 2013

a(35) > 10^6. - Ryan Propper, Jul 22 2015

References

• 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]
• 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]
• 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]

Links

• Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010. See pp.79-80.
• 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

3^n - 2 = A058481(n).

Cf. A058959, A171271. - M. F. Hasler and Farideh Firoozbakht, Dec 07 2009

Programs

Mathematica

A014224 = {}; Do[If[PrimeQ[3^n - 2], Print[n]; AppendTo[A014224, n]], {n, 10^5}]; A014224 (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Do[If[Mod[n, 4] != 3 && PrimeQ[3^n - 2], Print[n]], {n, 10000}] (* M. F. Hasler and Farideh Firoozbakht, Dec 07 2009 *)

PARI

for(n=1,1e4,if(ispseudoprime(3^n-2),print1(n", "))) \\ Charles R Greathouse IV, Jul 19 2011

Extensions

Corrected by Andrey V. Kulsha, Feb 04 2001

a(26) = 19217, a(27) = 20746 from Ryan Propper, May 11 2007

a(28) = 31722 from Henri Lifchitz, Oct 2002

a(29) = 37056 from Henri Lifchitz, Oct 2004

a(30) = 69581 from Henri Lifchitz, Jan 2005

a(31) = 195430 from Theodore Burton, Feb 2007

a(32) = 225922 from Ryan Propper, Aug 18 2013

a(33) = 506233 from Ryan Propper, Sep 02 2013

a(34) = 761457 from Ryan Propper, Jul 22 2015

a(35) = 1180181 from Jorge Coveiro, May 22 2020