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A014224
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Numbers n such that 3^n - 2 is prime.
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51
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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
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OFFSET
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1,1
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COMMENTS
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Contribution from M. F. Hasler and Farideh Firoozbakht, Dec 07 2009: (Start)
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 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. (End)
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
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REFERENCES
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Daniel Minoli, 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]
Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134) [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
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]
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LINKS
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Table of n, a(n) for n=1..31.
Henri & Renaud Lifchitz, PRP Records.
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MATHEMATICA
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lst={}; Do[If[PrimeQ[3^n-2], Print[n]; AppendTo[lst, n]], {n, 10^5}]; lst [From Vladimir Joseph Stephan Orlovsky, Aug 21 2008]
Do[If[Mod[n, 4]!=3 && PrimeQ[3^n-2], Print[n]], {n, 10000}] [From M. F. Hasler and Farideh Firoozbakht, Dec 07 2009]
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PROG
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(PARI) for(n=1, 1e4, if(ispseudoprime(3^n-2), print1(n", "))) \\ Charles R Greathouse IV, Jul 19 2011
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CROSSREFS
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3^n - 2 = A058481(n).
Cf. A058959, A171271. [From 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
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KEYWORD
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nonn,changed
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AUTHOR
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Jud McCranie
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EXTENSIONS
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Corrected by Andrey V. Kulsha, Feb 04 2001.
a(26)=19217, a(27)=20746 from Ryan Propper (rpropper(AT)stanford.edu), 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
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STATUS
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approved
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