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A055025 Norms of Gaussian primes. 19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is the range of the norm a^2 + b^2 of Gaussian primes a + b i. A239361 lists for each norm value a(n) one of the Gaussian primes as a, b with a >= b >= 0. In A239397 a >= 0 and b >= 0, beginning with 1, 1 for a(1)= 2, and then a(n) appears for a, b as well as for the following b, a, with a > b. - Wolfdieter Lang, Mar 24 2014

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

REFERENCES

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.

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

Eric Weisstein's World of Mathematics, Gaussian prime

Wikipedia, Table of Gaussian integer factorizations

Index entries for Gaussian integers and primes

FORMULA

Consists of 2; rational primes = 1 (mod 4) [A002144]; and squares of rational primes = 3 (mod 4) [A002145^2].

EXAMPLE

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.

MATHEMATICA

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 *)

CROSSREFS

Cf. A055026-A055029, A055664-A055666.

Cf. A001481.

Cf. A239397, A239621 (Gaussian primes).

Sequence in context: A130244 A161569 A182814 * A178805 A088907 A130235

Adjacent sequences:  A055022 A055023 A055024 * A055026 A055027 A055028

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Jun 09 2000

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

STATUS

approved

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Last modified October 21 13:08 EDT 2014. Contains 248377 sequences.