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A014224 Numbers n such that 3^n - 2 is prime. 53
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

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

Table of n, a(n) for n=1..33.

Henri & Renaud Lifchitz, PRP Records.

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]

PROG

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

CROSSREFS

3^n - 2 = A058481(n).

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

Sequence in context: A003306 A136585 A122721 * A175342 A077312 A140779

Adjacent sequences:  A014221 A014222 A014223 * A014225 A014226 A014227

KEYWORD

nonn

AUTHOR

Jud McCranie

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

STATUS

approved

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Last modified November 21 06:24 EST 2014. Contains 249769 sequences.