

A055025


Norms of Gaussian primes.


23



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

1,1


COMMENTS

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 JeanChristophe 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 indexn 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 EisensteinJacobi 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.  JeanChristophe 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].
a(n) ~ 2n log n.  Charles R Greathouse IV, Feb 06 2017


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]] (* JeanFrançois 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] (* JeanFrançois Alcover, Dec 07 2012 *)


PROG

(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


CROSSREFS

Cf. A055026A055029, A055664A055666, A001481.
Cf. A239397, A239621 (Gaussian primes).
Sequence in context: A161569 A263086 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



