This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346721 #36 Feb 16 2025 08:34:02 %S A346721 0,2,3,5,6,11,17,19,21,23,28,30,32,45,47,57,59,61,63,65,67,69,76,78, %T A346721 80,82,84,103,107,121,125,127,129,131,135,137,139,148,150,152,156,158, %U A346721 160,187,189,211,213 %N A346721 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-7)), as explained in the comments. Number the cells along the counterclockwise hexagonal spiral that starts with cells 0 and 1 representing integers 0 and 1. List the cells that represent 0 or a prime in the ring. %C A346721 In this entry we use "rational integers" to refer to integers in their usual sense as whole numbers - they form a subset of the algebraic integers that form the ring, which we denote "R". %C A346721 The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-7) or z = (x+0.5) + (y+0.5)*sqrt(-7) where x and y are rational integers. Plotted as points on a plane, they can be joined in a grid of isosceles triangles or be seen as the center points of hexagonal regions. When the latter are adjusted to make them regular, it makes for appealing diagrams, which we will come to shortly. %C A346721 (To be precise, we map each element, z, to the region of the complex plane containing the points that have z as their nearest ring element, then map these (hexagonal) regions continuously to the cells of a (regular) hexagonal grid.) %C A346721 R is one of 9 related rings that are unique factorization domains, meaning their elements factorize into prime elements in a unique way, just as with rational integers and prime numbers. See the Wikipedia link or the Stark reference, for example. %C A346721 This set of sequences is inspired by tilings: see the Wichmann link. Each tiling represents one of the 9 rings and shows the primes as distinctively colored squares or hexagons as appropriate. %C A346721 General properties of the related hexagonal spiral sequences: (Start) %C A346721 R is one of 7 rings where hexagons are appropriate. Each has elements of the form x + y*sqrt(-p) and (x+0.5) + (y+0.5)*sqrt(-p), where p is a (rational) prime congruent to 3 modulo 4. %C A346721 When mapping the grid cells to quadratic integers, it is often convenient to write the latter as a + w*b, where w = 0.5*(1+sqrt(-p)). Cell m on the spiral represents A307011(m) + w*A307012(m). %C A346721 We can find the primes without advanced mathematics, using multiplication formulas and a sieve as explained below. %C A346721 w^2 = w - c, where c = (p+1)/4 (which is an integer as p == 3 (mod 4)). So, in general, the product of a_1 + w*b_1 and a_2 + w*b_2 is (a_1*a_2 - c*b_1*b_2) + w*(a_1*b_2 + a_2*b_1 + b_1*b_2). The norm (absolute square) of a + w*b is a^2 + a*b + c*b^2. %C A346721 For k >= 1, the algebraic integers represented by cells numbered 3k*(k-1)+1 to 3k*(k+1) on the spiral (cells A003215(k-1) to A028896(k)) are positioned along a hexagon in the complex plane; they include rational integers k and -k, and have norms in the range [k^2*(4c-1)/4c, k^2*c] = [k^2*p/(p+1), k^2*c]. %C A346721 To determine the primes we may list the ring elements in an order such that they have nondecreasing norm, and use a sieve to remove the products of nonunits. So, we are only interested in elements with norm greater than 1 (i.e. nonzero, nonunit). At each round of sieving we note the first element, z, whose products we have not yet removed, and remove in turn the product of z and each element from z onwards in the list. %C A346721 (End) %D A346721 L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910. %D A346721 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 A346721 OEIS Wiki, <a href="http://oeis.org/wiki/Algebraic_integers">Algebraic integers</a>. %H A346721 OEIS Wiki, <a href="https://oeis.org/wiki/Norm">Norm</a>. %H A346721 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbsoluteSquare.html">Absolute Square</a>, <a href="https://mathworld.wolfram.com/ComplexPlane.html">Complex Plane</a>, <a href="https://mathworld.wolfram.com/HexagonalGrid.html">Hexagonal Grid</a>, <a href="https://mathworld.wolfram.com/RingofIntegers.html">Ring of Integers</a>. %H A346721 Brian Wichmann, <a href="http://www.tilingsearch.org/special/ufd.pdf">Tiling for Unique Factorization Domains</a>, Jul 22 2019. See Figure 5. %H A346721 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quadratic_integer">Quadratic integer</a>. %H A346721 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unique_factorization_domain">Unique factorization domain</a>. %F A346721 m is a term if and only if A345764(m) is a term. %e A346721 Table showing derivation of initial terms. %e A346721 The ring element, z, represented by spiral cell m is shown in the form A307011(m) + A307012(m)*w, where w = 0.5*(1+sqrt(-7)). %e A346721 The column headed "(x,y)" gives x and y when z is written in the form z = x + y*sqrt(-7). %e A346721 A307011(m) %e A346721 | A307012(m) %e A346721 m | | z (x,y) status n a(n)=m %e A346721 | | %e A346721 0 0 0 0 ( 0.0, 0.0) zero 1 0 %e A346721 1 1 0 1 ( 1.0, 0.0) unit %e A346721 2 0 1 w ( 0.5, 0.5) prime 2 2 %e A346721 3 -1 1 -1+w (-0.5, 0.5) prime 3 3 %e A346721 4 -1 0 -1 (-1.0, 0.0) unit %e A346721 5 0 -1 -w (-0.5,-0.5) prime 4 5 %e A346721 6 1 -1 1-w ( 0.5,-0.5) prime 5 6 %e A346721 7 2 -1 2-w ( 1.5,-0.5) = -w*w %e A346721 8 2 0 2 ( 2.0, 0.0) = (1-w)*w %e A346721 9 1 1 1+w ( 1.5, 0.5) = (1-w)*(w-1) %e A346721 10 0 2 2w ( 1.0, 1.0) = 2*w %e A346721 11 -1 2 -1+2w ( 0.0, 1.0) prime 6 11 %Y A346721 Cf. A003173. %Y A346721 Norms of primes in R: A090348. %Y A346721 Sequences related to the geometry of the spiral: A003215, A028896, A307011, A307012, A345764. %Y A346721 Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346722 (D=-11), A346723 (D=-19), A346724 (D=-43), A346725 (D=-67), A346726 (D=-163). %K A346721 nonn,more %O A346721 1,2 %A A346721 _Peter Munn_, Jul 30 2021