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A055025
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Norms of Gaussian primes.
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24
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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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This is the range of the norm a^2 + b^2 of Gaussian primes a + b i. A239621 lists for each norm value a(n) one of the Gaussian primes as a, b with a >= b >= 0. In A239397, any of these (a, b) is followed by (b, a), except for a = b = 1. - Wolfdieter Lang, Mar 24 2014, edited by M. F. Hasler, Mar 09 2018
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)
The sequence is formed of 2, the prime numbers of form 4k+1, and the square of other primes (of form 4k+3). These are the primitive elements of A001481. With 0 and 1, they are the numbers that are uniquely decomposable in the sum of two squares. - Jean-Christophe Hervé, Nov 17 2013
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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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Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Gaussian prime
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].
a(n) ~ 2n log n. - Charles R Greathouse IV, Feb 06 2017
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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). In A239621 2+i is listed as 2, 1.
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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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PROG
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(PARI) list(lim)=my(v=List()); if(lim>=2, listput(v, 2)); forprime(p=3, sqrtint(lim\1), if(p%4==3, listput(v, p^2))); forprime(p=5, lim, if(p%4==1, listput(v, p))); Set(v) \\ Charles R Greathouse IV, Feb 06 2017
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CROSSREFS
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Cf. A055026-A055029, A055664-A055666, A001481.
Cf. A239397, A239621 (Gaussian primes).
Sequence in context: A161569 A263086 A182814 * A178805 A088907 A130235
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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