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A051783
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Numbers n such that 3^n + 2 is prime.
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17
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0, 1, 2, 3, 4, 8, 10, 14, 15, 24, 26, 36, 63, 98, 110, 123, 126, 139, 235, 243, 315, 363, 386, 391, 494, 1131, 1220, 1503, 1858, 4346, 6958, 7203, 10988, 22316, 33508, 43791, 45535, 61840, 95504, 101404, 106143, 107450, 136244, 178428
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| (Contribution from F. Firoozbakht (mymontain(AT)yahoo.com) and M. F. Hasler (www.univ-ag.fr/~mhasler), Dec 06 2009, Start)
If Q is a perfect number such that gcd(Q,3(3^a(n)+2))=1, then x=3^(a(n)-1)*(3^a(n)+2)*Q is a solution of the equation sigma(x)=3(x-Q). This is a result of the following theorem:
Theorem: If Q is a (q-1)-perfect number for some prime q, then for all integers t, the equation sigma(x)=q*x-(2t+1)*Q has the solution x=q^(k-1)*p*Q whenever k is a positive integer such that p=q^k+2t is prime, gcd(q^(k-1), p)=1 and gcd(q^(k-1)*p,Q)=1.
Note that by taking t=-1/2(m*q+1), this theorem gives us some solutions of the equation sigma(x)=q*(x+m*Q). See comment lines of the sequence A058959. (End)
No further terms < 200000. - Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 31 2011
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LINKS
| Henri & Renaud Lifchitz, PRP Records.
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EXAMPLE
| 3^8+2 = 6563 is prime, so 8 is in the sequence.
3^26 + 2 = 2541865828331, a prime number
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MATHEMATICA
| Select[Range[0, 20000], PrimeQ[3^# + 2] &]
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CROSSREFS
| Cf. A057735, A087885, A014224, A058959.
Sequence in context: A061193 A180941 A135772 * A033083 A005542 A037171
Adjacent sequences: A051780 A051781 A051782 * A051784 A051785 A051786
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KEYWORD
| nonn,hard
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AUTHOR
| Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Dec 09 1999
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EXTENSIONS
| {4346, 6958, 7203} from Dave Rusin (rusin(AT)math.niu.edu), Sep 29 2000
10988 from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 21 2004
{22316, 33508} found by Henry Lifchitz, Sep-Oct 2002
{43791, 45535, 61840} found by Henry Lifchitz, Oct-Nov 2004
95504 found by Wojciech Florek Dec 15 2005. - Alexander Adamchuk (alex(AT)kolmogorov.com), Mar 02 2008
Edited by N. J. A. Sloane, Dec 19 2009
{101404, 106143, 107450, 136244} from Mike Oakes, Nov 2009
178428 from Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 29 2011
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