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A222008
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Primes of the form 4^k + 1 for some k > 0.
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3
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OFFSET
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1,1
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COMMENTS
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Same as Fermat primes 2^(2^m) + 1 for m >= 1. See A019434 for comments, etc.
Chebyshev showed that 3 is a primitive root of all primes of the form 2^(2*k) + 1 with k > 0. If the sequence is infinite, then Artin's conjecture ("every nonsquare integer n != -1 is a primitive root of infinitely many primes q") is true for n = 3.
As a(n) is a Fermat prime > 3, by Pépin's test a(n) has primitive root 3.
Conjecture: let p a prime number, a(n) is not congruent to p mod (p^2-3)/2. - Vincenzo Librandi, Jun 15 2014
This conjecture is false when p = a(n), but may be true for primes p != a(n). - Jonathan Sondow, Jun 15 2014
Primes p with the property that k-th smallest divisor of its squares p^2, for all 1 <= k <= tau(p^2), contains exactly k "1" digits in its binary representation. Corresponding values of squares p^2: 25, 289, 66049, 4295098369. Example: p = 257, set of divisors of p^2 = 66049 in binary representation: {1, 100000001, 10000001000000001}. Subsequence of A255401. - Jaroslav Krizek, Dec 21 2016
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REFERENCES
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Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 102, nr. 3.
P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.
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LINKS
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Table of n, a(n) for n=1..4.
P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, p. 306-313.
R. Fueter, Über primitive Wurzeln von Primzahlen, Comment. Math. Helv., 18 (1946), 217-223, p. 217.
Wikipedia, Pépin's test
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root
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FORMULA
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a(n) = A019434(n+1) for n > 0.
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EXAMPLE
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4^1 + 1 = 5 is prime, so a(1) = 5. Also, 3^k == 3, 4, 2, 1 (mod 5) for k = 1, 2, 3, 4, resp., so 3 is a primitive root for a(1).
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MATHEMATICA
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Select[Table[4^k + 1, {k, 10^3}], PrimeQ] (* Michael De Vlieger, Dec 22 2016 *)
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CROSSREFS
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Cf. A005596, A019334, A019434, A090866, A221981.
Sequence in context: A273948 A271657 A273999 * A274000 A093428 A274002
Adjacent sequences: A222005 A222006 A222007 * A222009 A222010 A222011
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KEYWORD
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nonn
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AUTHOR
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Jonathan Sondow, Feb 04 2013
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STATUS
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approved
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