

A014224


Numbers k such that 3^k  2 is prime.


63



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


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



MATHEMATICA



PROG



CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS

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


STATUS

approved



