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A234310
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Primes of the form 4^k + 4^m - 1, where k and m are positive integers.
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14
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7, 19, 31, 67, 79, 127, 271, 1039, 1087, 1279, 4099, 4111, 4159, 5119, 8191, 16447, 20479, 65539, 65551, 65599, 81919, 131071, 262147, 262399, 263167, 266239, 524287, 1049599, 1114111, 1310719, 4194319, 4194559, 4195327, 16842751, 17825791, 67108879
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OFFSET
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1,1
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COMMENTS
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Clearly each term is congruent to 1 modulo 6.
By the conjecture in A234309, this sequence should have infinitely many terms.
Note that any Mersenne prime greater than 3 has the form 2^{2*k+1} - 1 = 4^k + 4^k - 1, where k is a positive integer.
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LINKS
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EXAMPLE
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a(1) = 7 since 7 = 4^1 + 4^1 - 1 is prime.
a(2) = 19 since 19 = 4^1 + 4^2 - 1 is prime.
a(3) = 31 since 31 = 4^2 + 4^2 - 1 is prime.
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MATHEMATICA
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n=0; Do[If[PrimeQ[4^k+4^m-1], n=n+1; Print[n, " ", 4^m+4^k-1]], {m, 1, 250}, {k, 1, m}]
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PROG
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(PARI) for(k=1, 30, for(m=1, k, if(ispseudoprime(t=4^k+4^m-1), print1(t", ")))) \\ Charles R Greathouse IV, Dec 23 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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