

A239621


Gaussian primes x + i*y, with x = a(2n1) >= y = a(2n) >= 0, sorted by norm.


5



1, 1, 2, 1, 3, 0, 3, 2, 4, 1, 5, 2, 6, 1, 5, 4, 7, 0, 7, 2, 6, 5, 8, 3, 8, 5, 9, 4, 10, 1, 10, 3, 8, 7, 11, 0, 11, 4, 10, 7, 11, 6, 13, 2, 10, 9, 12, 7, 14, 1, 15, 2, 13, 8, 15, 4, 16, 1, 13, 10, 14, 9, 16, 5, 17, 2, 13, 12, 14, 11, 16, 9, 18, 5, 17, 8, 19, 0
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OFFSET

1,3


COMMENTS

The condition a >= b >= 0 implies that there is only one Gaussian prime for each norm.  T. D. Noe, Mar 26 2014
The real parts and imaginary parts are listed as a(2n1) = A300587(n) and a(2n) = A300588(n), respectively. Sequence A239397 lists the pair (y, x) after each pair (x, y), except for (1, 1).  M. F. Hasler, Mar 10 2018


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10106 (5053 pairs)
Eric Weisstein's World of Mathematics, Gaussian prime
Wikipedia, Complex Number


EXAMPLE

From M. F. Hasler, Mar 09 2018: (Start)
Sorted by norm, the smallest Gaussian primes z = x + iy in the first halfquadrant x >= y >= 0 are:
a(1) + i*a(2) = 1 + i;
a(3) + i*a(4) = 2 + i;
a(5) + i*a(6) = 3;
... (End)


MATHEMATICA

mx = 20; lst = Flatten[Table[{a, b}, {a, 0, mx}, {b, 0, a}], 1]; qq = Select[lst, Norm[#] <= mx && PrimeQ[#[[1]] + I*#[[2]], GaussianIntegers > True] &]; Sort[qq, Norm[#1] < Norm[#2] &]


PROG

(PARI) {for(n=2, 400, f=factor(n*I)/*factor in Z[i]*/; matsize(f)[1]<=2 && vecsum(f[, 2])==2+(f[1, 1]==I) /*either I*p^2 or w*conj(w/I), maybe (1+I)^2 */ && printf("%d, ", vecsort([real(f=f[3f[1, 2], 1]), imag(f)], , 4)))} \\ For illustrative use.  M. F. Hasler, Mar 09 2018


CROSSREFS

Cf. A055025 (norms of Gaussian primes), A239397.
Cf. A300587, A300588.
Sequence in context: A122170 A066029 A141198 * A231204 A180987 A092093
Adjacent sequences: A239618 A239619 A239620 * A239622 A239623 A239624


KEYWORD

nonn


AUTHOR

T. D. Noe, Mar 22 2014


EXTENSIONS

Name changed and in cf. complex > Gaussian  Wolfdieter Lang, Mar 25 2014
Name edited by M. F. Hasler, Mar 09 2018


STATUS

approved



