A345437 - OEIS

%I #36 Dec 16 2021 20:13:17

%S 0,2,3,4,6,7,8,25,26,28,29,32,34,37,38,40,41,44,46,57,63,73,79

%N Represent the ring R = {x+y*sqrt(-2): x, y rational integers} by the cells centered at the points (x,y) of a square grid; number the cells of the grid along a counterclockwise square spiral, with the cells at (0,0) and (1,0) numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in R.

%C R is the ring of integers in the quadratic number field Q(sqrt(-2)). The element x+y*sqrt(-2) in R has norm x^2+2*y^2.

%C A033715 gives the number of elements in R with norm n.

%C There are two units, +-1, of norm 1.

%C A341784 gives the norms of the primes in R, and A345438 gives the numbers of primes of those norms.

%D H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

%H N. J. A. Sloane, <a href="/A345437/a345437.pdf">Illustration of initial terms</a> [An enlargement of Figure 3 of Wichmann (2019), showing the numbering of the initial cells of the square spiral. The origin is black, the two units +-1 are red, and the primes are blue.]

%H Brian Wichmann, <a href="http://www.tilingsearch.org/special/ufd.pdf">Tiling for Unique Factorization Domains</a>, Jul 22 2019

%H Brian Wichmann, <a href="/A345437/a345437.png">Detail of Figure 3 from the previous link</a>

%e One can read off the primes from the blue cells in the illustration. The first few primes are +-sqrt(-2), 2 of norm 2; +-1+-sqrt(-2), 4 of norm 3; +-3+-sqrt(-2), 4 of norm 11; ... (see A345438).

%Y Cf. A003173, A033715, A341784, A345435, A345436, A345438.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jun 23 2021