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A051783
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Numbers k such that 3^k + 2 is prime.
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47
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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, 361608, 504206, 1753088
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OFFSET
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1,3
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COMMENTS
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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)
a(45) and a(46) are probable primes because a primality certificate has not yet been found. They have been verified PRP with mprime. - Luke W. Richards, May 04 2018
Conjecture: The number n = 3^k + 2 is prime if and only if 2^((n-1)/2) == -1 (mod n). - Maheswara Rao Valluri, Jun 01 2020. [Note that this is an if and only if assertion, so it does not follow from Fermat's Little Theorem. - N. J. A. Sloane, Sep 07 2020]
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LINKS
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EXAMPLE
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3^8 + 2 = 6563 is prime, so 8 is in the sequence.
3^26 + 2 = 2541865828331, a prime number, so 26 is in the sequence.
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MATHEMATICA
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A051783 = Select[Range[0, 20000], PrimeQ[3^# + 2] &]
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PROG
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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EXTENSIONS
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{101404, 106143, 107450, 136244} from Mike Oakes, Nov 2009
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STATUS
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approved
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