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A234310
Primes of the form 4^k + 4^m - 1, where k and m are positive integers.
14
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
OFFSET
1,1
COMMENTS
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.
LINKS
EXAMPLE
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.
MATHEMATICA
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}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 23 2013
STATUS
approved