A345437 -id:A345437 - cp's OEIS frontend

A345440 Represent the ring R = {x+ysqrt(-2): x, y rational integers} by the cells centered at the points (x,y) of a square grid, as in A345437; number the cells of the grid along a counterclockwise square spiral, with the cells at (0,0) and (1,0) numbered 0, 1; then a(n) is the norm x^2+2y^2 of the element of R occupying the n-th cell.

0, 1, 3, 2, 3, 1, 3, 2, 3, 6, 4, 6, 12, 9, 8, 9, 12, 6, 4, 6, 12, 9, 8, 9, 12, 17, 11, 9, 11, 17, 27, 22, 19, 18, 19, 22, 27, 17, 11, 9, 11, 17, 27, 22, 19, 18, 19, 22, 27, 34, 24, 18, 16, 18, 24, 34, 48, 41, 36, 33, 32, 33, 36, 41, 48, 34, 24, 18, 16, 18, 24

Offset: 0

N. J. A. Sloane, Jun 25 2021

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A345440
N. J. A. Sloane, Illustration of initial terms [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.]
N. J. A. Sloane, Illustration of initial terms. [The cell numbers are black, their norms are red.]
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019.

Cf. A174344, A274923, A336336, A345435, A345437, A345439.

PARI
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See Links section.
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a(n) = A174344(n+1)^2 + 2*A274923(n+1)^2. - Rémy Sigrist, Jun 26 2021

More terms from Rémy Sigrist, Jun 26 2021

A345435 Represent the ring of Eisenstein integers E = {x+yomega: x, y rational integers, omega = exp(2Pi*i/3)} by the cells of a hexagonal grid; number the cells of the grid along a counterclockwise hexagonal spiral, with the cells 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

Original entry on oeis.org

0, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 90, 91, 95, 97, 101, 103, 107, 109, 113, 115

Offset: 1

N. J. A. Sloane, Jun 23 2021

nonn

The Eisenstein integer represented by cell m is A307013(m) + A307012(m)*omega. Thus the set of Eisenstein primes is {A307013(a(n)) + A307012(a(n))*omega : n >= 2}. - Peter Munn, Jun 26 2021
The Eisenstein integer a + b*omega has norm a^2 - a*b + b^2 (see A003136). The number of Eisenstein integers of norm n is given by A004016(n).
The norms of the Eisenstein primes are given in A055664, and the number of Eisenstein primes of norm n is given in A055667.
Reid's 1910 book (still in print) is still the best reference for the Eisenstein integers and similar rings.

			The smallest Eisenstein integers are 0 (of norm 0), and the six units of norm 1, namely (writing w for omega) +-1, +-w, +-w^2.
The first few Eisenstein primes are (here u is any of the six units):
   u*(2+w), norm = 3, number = 6;
   2*u, norm = 4, number = 6;
   u*(3+w), norm = 7, number = 6;
   u*(3+2*w), norm = 7, number = 6 (so there are 12 primes of norm 7 - see A055667).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.4, p. 111.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.
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.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A345435
N. J. A. Sloane, Illustration of initial terms [An enlargement of Figure 1 of Wichmann (2019), showing the numbering of the initial cells of the hexagonal spiral.]
Eric Weisstein's World of Mathematics, Eisenstein Prime
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019.
Brian Wichmann, The Eisenstein integers, with the primes shaded [Figure 1 from the previous link]

Cf. A003136, A003173, A003627, A004016, A007645, A055664, A055667, A307012, A307013, A308412, A345436, A345437.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Jun 26 2021

A345436 Represent the ring of Gaussian integers E = {x+y*i: x, y rational integers, i = sqrt(-1)} by the cells of a square grid; number the cells of the grid along a counterclockwise square spiral, with the cells representing the ring identities 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

Original entry on oeis.org

0, 2, 4, 6, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 59, 61, 67, 69, 75, 77, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157

Offset: 1

N. J. A. Sloane, Jun 23 2021

nonn

The cell with spiral index m represents the Gaussian integer A174344(m+1) + A274923(m+1) * i. So the set of Gaussian primes is {A174344(a(n)+1) + A274923(a(n)+1) * i : n >= 2}. - Peter Munn, Aug 02 2021
The Gaussian integer z = x+i*y has norm x^2+y^2. There are four units (of norm 1), +-1, +-i. The number of Gaussian integers of norm n is A004018(n).
The norms of the Gaussian primes are listed in A055025, and the number of primes with a given norm is given in A055026.
The successive norms of the Gaussian integers along the square spiral are listed in A336336.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.2, p. 106.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.
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.

Eric Weisstein's World of Mathematics, Gaussian prime.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Fig. 2.

Equals A308412 - 1. Cf. A345435, A345437.

Cf. also A003173, A004018, A055025, A055026, A103431, A174344, A274923, A336336.

Name clarified by Peter Munn, Aug 02 2021

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.

Original entry on oeis.org

0, 2, 3, 5, 6, 11, 17, 19, 21, 23, 28, 30, 32, 45, 47, 57, 59, 61, 63, 65, 67, 69, 76, 78, 80, 82, 84, 103, 107, 121, 125, 127, 129, 131, 135, 137, 139, 148, 150, 152, 156, 158, 160, 187, 189, 211, 213

Offset: 1

Peter Munn, Jul 30 2021

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".
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.
(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.)
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.
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.
General properties of the related hexagonal spiral sequences: (Start)
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.
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).
We can find the primes without advanced mathematics, using multiplication formulas and a sieve as explained below.
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.
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].
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.
(End)

			Table showing derivation of initial terms.
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)).
The column headed "(x,y)" gives x and y when z is written in the form z = x + y*sqrt(-7).
      A307011(m)
       |  A307012(m)
   m   |   |      z      (x,y)      status         n  a(n)=m
       |   |
   0   0   0      0   ( 0.0, 0.0)     zero         1    0
   1   1   0      1   ( 1.0, 0.0)     unit
   2   0   1      w   ( 0.5, 0.5)    prime         2    2
   3  -1   1   -1+w   (-0.5, 0.5)    prime         3    3
   4  -1   0     -1   (-1.0, 0.0)     unit
   5   0  -1     -w   (-0.5,-0.5)    prime         4    5
   6   1  -1    1-w   ( 0.5,-0.5)    prime         5    6
   7   2  -1    2-w   ( 1.5,-0.5)  =    -w*w
   8   2   0      2   ( 2.0, 0.0)  = (1-w)*w
   9   1   1    1+w   ( 1.5, 0.5)  = (1-w)*(w-1)
  10   0   2     2w   ( 1.0, 1.0)  =     2*w
  11  -1   2  -1+2w   ( 0.0, 1.0)    prime         6   11

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
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.

OEIS Wiki, Algebraic integers.
OEIS Wiki, Norm.
Eric Weisstein's World of Mathematics, Absolute Square, Complex Plane, Hexagonal Grid, Ring of Integers.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 5.
Wikipedia, Quadratic integer.
Wikipedia, Unique factorization domain.

Cf. A003173.
Norms of primes in R: A090348.
Sequences related to the geometry of the spiral: A003215, A028896, A307011, A307012, A345764.
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).

m is a term if and only if A345764(m) is a term.

A346722 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-11)) 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.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 37, 43, 49, 55, 61, 62, 64, 66, 68, 69, 71, 74, 76, 77, 79, 81, 83, 84, 86, 89, 103, 107, 121, 125, 128, 131, 133, 135, 138, 142, 145, 149, 152, 154, 156, 159, 163, 166, 173, 175, 177, 179, 197, 199, 201, 203

Offset: 1

Peter Munn, Jul 30 2021

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".
The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-11) or z = (x+0.5) + (y+0.5)*sqrt(-11) 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. Adjusting the regions to be regular hexagons makes for appealing diagrams, which we will come to shortly.
(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.)
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.
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.
6 other rings (of the 9) can be mapped to the hexagonal grid in the same way. See the comments entitled "General properties of the related hexagonal spiral sequences" in A346721.

			The sequence is constructed in the same way as A346721, but the relevant prime is 11 instead of 7. See the example section of A346721.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
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.

OEIS Wiki, Algebraic integers.
Eric Weisstein's World of Mathematics, Complex Plane, Hexagonal Grid, Ring of Integers.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 6.
Wikipedia, Quadratic integer.
Wikipedia, Unique factorization domain.

Cf. A003173, A296920, A345764.
Norms of primes in R: A341785.
Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346721 (D=-7), A346723 (D=-19), A346724 (D=-43), A346725 (D=-67), A346726 (D=-163).

m is a term if and only if A345764(m) is a term.

A346723 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-19)) 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.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 34, 35, 39, 41, 51, 53, 91, 101, 109, 119, 128, 129, 130, 132, 134, 136, 137, 138, 141, 143, 144, 146, 149, 150, 151, 153, 155, 157, 158, 159, 162, 164, 165, 167, 171, 175, 177

Offset: 1

Peter Munn, Aug 13 2021

nonn

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".
The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-19) or z = (x+0.5) + (y+0.5)*sqrt(-19) 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. Adjusting the regions to be regular hexagons makes for appealing diagrams, which we will come to shortly.
(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.)
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.
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.
6 other rings (of the 9) can be mapped to the hexagonal grid in the same way. See the comments entitled "General properties of the related hexagonal spiral sequences" in A346721.

			The sequence is constructed in the same way as A346721, but the relevant prime is 19 instead of 7. See the example section of A346721.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
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.

OEIS Wiki, Algebraic integers.
Eric Weisstein's World of Mathematics, Complex Plane, Hexagonal Grid, Ring of Integers.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 7.
Wikipedia, Quadratic integer.
Wikipedia, Unique factorization domain.

Cf. A003173, A345764.
Norms of primes in R: A341787.
Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346721 (D=-7), A346722 (D=-11), A346724 (D=-43), A346725 (D=-67), A346726 (D=-163).

m is a term if and only if A345764(m) is a term.

A346724 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-43)) 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.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 91, 95, 97, 101, 109

Offset: 1

Peter Munn, Aug 23 2021

nonn

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".
The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-43) or z = (x+0.5) + (y+0.5)*sqrt(-43) 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. Adjusting the regions to be regular hexagons makes for appealing diagrams, which we will come to shortly.
(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.)
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.
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.
6 other rings (of the 9) can be mapped to the hexagonal grid in the same way. See the comments entitled "General properties of the related hexagonal spiral sequences" in A346721.

			The sequence is constructed in the same way as A346721, but the relevant prime is 43 instead of 7. See the example section of A346721.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
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.

OEIS Wiki, Algebraic integers.
Eric Weisstein's World of Mathematics, Complex Plane, Hexagonal Grid, Ring of Integers.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 8.
Wikipedia, Quadratic integer.
Wikipedia, Unique factorization domain.

Cf. A003173, A345764.
Norms of primes in R: A341788.
Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346721 (D=-7), A346722 (D=-11), A346723 (D=-19), A346725 (D=-67), A346726 (D=-163).

m is a term if and only if A345764(m) is a term.

A346725 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-67)) 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.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 91, 95, 97, 101, 103, 107, 109

Offset: 1

Peter Munn, Aug 23 2021

nonn

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".
The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-67) or z = (x+0.5) + (y+0.5)*sqrt(-67) 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. Adjusting the regions to be regular hexagons makes for appealing diagrams, which we will come to shortly.
(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.)
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.
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.
6 other rings (of the 9) can be mapped to the hexagonal grid in the same way. See the comments entitled "General properties of the related hexagonal spiral sequences" in A346721.
This sequence first differs from A346724 at a(70) = 103 <> A346724(70) = 109 and from A346726 at a(116) = 171 <> A346726(116) = 169.
The first rational prime not to appear as a cell number in this sequence is 191.

			The sequence is constructed in the same way as A346721, but the relevant prime is 67 instead of 7. See the example section of A346721.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
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.

OEIS Wiki, Algebraic integers.
Eric Weisstein's World of Mathematics, Complex Plane, Hexagonal Grid, Ring of Integers.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 9.
Wikipedia, Quadratic integer.
Wikipedia, Unique factorization domain.

Cf. A003173, A345764.
Norms of primes in R: A341789.
Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346721 (D=-7), A346722 (D=-11), A346723 (D=-19), A346724 (D=-43), A346726 (D=-163).

m is a term if and only if A345764(m) is a term.

A346726 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-163)) 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.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 91, 95, 97, 101, 103, 107, 109

Offset: 1

Peter Munn, Aug 23 2021

nonn

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".
The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-163) or z = (x+0.5) + (y+0.5)*sqrt(-163) 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. Adjusting the regions to be regular hexagons makes for appealing diagrams, which we will come to shortly.
(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.)
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.
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.
6 other rings (of the 9) can be mapped to the hexagonal grid in the same way. See the comments entitled "General properties of the related hexagonal spiral sequences" in A346721.
This sequence first differs from A346724 at a(70) = 103 <> A346724(70) = 109 and from A346725 at a(116) = 169 <> A346725(116) = 171.

			The sequence is constructed in the same way as A346721, but the relevant prime is 163 instead of 7. See the example section of A346721.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
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.

OEIS Wiki, Algebraic integers.
Eric Weisstein's World of Mathematics, Complex Plane, Hexagonal Grid, Ring of Integers.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 10.
Wikipedia, Quadratic integer.
Wikipedia, Unique factorization domain.

Cf. A003173, A345764.
Norms of primes in R: A341790.
Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346721 (D=-7), A346722 (D=-11), A346723 (D=-19), A346724 (D=-43), A346725 (D=-67).

m is a term if and only if A345764(m) is a term.

A345438 Number of primes in the ring R = {x+ysqrt(-2): x, y rational integers} with norm x^2+2y^2 equal to A341784(n).

Original entry on oeis.org

2, 4, 4, 4, 4, 2, 4, 4

Offset: 1

N. J. A. Sloane, Jun 24 2021

See A345437 for further information.

			The two primes in R with smallest norm are +-sqrt(-2), of norm 2 = A341784(1), so a(1) = 2.

Cf. A341784, A345437.

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A345438 Number of primes in the ring R = {x+ysqrt(-2): x, y rational integers} with norm x^2+2y^2 equal to A341784(n).