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A055025
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Norms of Gaussian primes.
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17
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2, 5, 9, 13, 17, 29, 37, 41, 49, 53, 61, 73, 89, 97, 101, 109, 113, 121, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 361, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 529, 541, 557, 569
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OFFSET
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1,1
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COMMENTS
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These are the primes in the ring of Gaussian integers a+bi, a and b rational integers, i = sqrt(-1).
From Jean-Christophe Hervé, May 01 2013: (Start)
The present sequence is related to the square lattice, and to its division in square sublattices. Let's say that an integer n divides a lattice if there exists a sublattice of index n. Example: 2, 4, 5 divide the square lattice. Then A001481 (norms of Gaussian integers) is the sequence of divisors of the square lattice. Say that n is a "prime divisor" if the index-n sublattice is not contained in any other sublattice except the original lattice itself. The present sequence gives the "prime divisors" of the square lattice.
Similarly, A055664 (Norms of Eisenstein-Jacobi primes) is the sequence of "prime divisors" of the hexagonal lattice. (End)
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Wikipedia, Table of Gaussian integer factorizations
Index entries for Gaussian integers and primes
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FORMULA
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Consists of 2; rational primes = 1 (mod 4) [A002144]; and squares of rational primes = 3 (mod 4) [A002145^2 ]
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EXAMPLE
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There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i).
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MATHEMATICA
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Union[(#*Conjugate[#] & )[ Select[Flatten[Table[a + b*I, {a, 0, 23}, {b, 0, 23}]], PrimeQ[#, GaussianIntegers -> True] & ]]][[1 ;; 55]] (* Jean-François Alcover, Apr 08 2011 *)
(* Or, from formula: *) maxNorm = 569; s1 = Select[Range[1, maxNorm, 4], PrimeQ]; s3 = Select[Range[3, Sqrt[maxNorm], 4], PrimeQ]^2; Union[{2}, s1, s3] (* Jean-François Alcover, Dec 07 2012 *)
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CROSSREFS
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Cf. A055026-A055029, A055664-...
Sequence in context: A130244 A161569 A182814 * A178805 A130235 A219647
Adjacent sequences: A055022 A055023 A055024 * A055026 A055027 A055028
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Jun 09 2000
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000
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STATUS
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approved
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