

A014224


Numbers n such that 3^n  2 is prime.


61



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
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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^(n1)*(3^n2) is a 2hyperperfect number, cf. A007593, and no other 2hyperperfect 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 MillerRabin 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. 277302. [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Daniel Minoli, Voice over MPLS, McGrawHill, New York, NY, 2002, ISBN 0071406158 (pp. 114134) [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..35.
Antal Bege, Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010. See pp.7980.
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
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^n2), 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: A250305 A136585 A122721 * A175342 A077312 A325558
Adjacent sequences: A014221 A014222 A014223 * A014225 A014226 A014227


KEYWORD

nonn,more


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
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


STATUS

approved



