A336336 -id:A336336 - cp's OEIS frontend

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.

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

A359058 a(n) = squared distance to the origin of the n-th vertex on a counterclockwise undulating spiral in a square grid.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 5, 4, 1, 2, 1, 4, 5, 2, 5, 4, 9, 10, 5, 8, 5, 10, 9, 16, 17, 10, 13, 8, 13, 10, 17, 16, 9, 10, 5, 8, 5, 10, 9, 16, 17, 10, 13, 8, 13, 10, 17, 16, 25, 26, 17, 20, 13, 18, 13, 20, 17, 26, 25, 36, 37, 26, 29, 20, 25, 18, 25, 20, 29, 26, 37, 36, 25, 26, 17, 20, 13, 18, 13, 20

Offset: 0

Hans G. Oberlack, Dec 14 2022

The spiral coordinates are A359216 and A359217.

			The spiral begins as follows and for instance point n=7 is at x=-2,y=1 so that a(7) = (-2)^2 + 1^2 = 5.
   y ^
     |
   4 |             17--16
     |              |   |
   3 |         13--10   9--10
     |          |           |
   2 |     13---8   5---4   5---8
     |      |       |   |       |
   1 | 17--10   5---2   1---2   5--10
     |  |       |           |       |
   0 | 16---9   4---1   0---1   4---9
     |      |       |           |       |
  -1 |     10---5   2---1   2---5  10--17
     |          |       |   |       |
  -2 |          8---5   4---5   8--13
     |              |           |
  -3 |             10---9  10--13
     |                  |   |
  -4 |                 16--17
     +------------------------------------>
       -4  -3  -2  -1   0   1   2   3   4 x

Rémy Sigrist, Table of n, a(n) for n = 0..10081
Hans G. Oberlack, Undulating counterclockwise spiral in a square grid
Rémy Sigrist, PARI program

Cf. A001481, A336336, A359216, A359217.

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a(n) = A359216(n)^2 + A359217(n)^2.

A345439 Consider the Eisenstein integers x + yomega, x and y rational integers, represented as the cells of an hexagonal grid; draw a hexagonal spiral as in A345435; a(n) is the norm x^2-xy+y^2 of the Eisenstein integer in the n-th cell of the spiral.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 7, 7, 9, 7, 7, 9, 7, 7, 9, 7, 7, 9, 7, 7, 9, 7, 7, 9, 13, 12, 13, 16, 13, 12, 13, 16, 13, 12, 13, 16, 13, 12, 13, 16, 13, 12, 13, 16, 13, 12, 13, 16, 21, 19, 19, 21, 25, 21, 19, 19, 21, 25, 21, 19, 19

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 A345439
N. J. A. Sloane, The start of the hexagonal spiral, showing the numbering of the cells. [An enlargement of Figure 1 of Wichmann (2019). The gray cells are the Eisenstein primes.]
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.
Brian Wichmann, The Eisenstein integers, with the primes shaded [Figure 1 from the previous link]

Cf. A336336, A345435.

PARI
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More terms from Rémy Sigrist, Jun 26 2021

A335298 a(n) is the squared distance between the points P(n) and P(0) on a plane, n >= 0, such that the distance between P(n) and P(n+1) is n+1 and, going from P(n) to P(n+2), a 90-degree left turn is taken in P(n+1).

Original entry on oeis.org

0, 1, 5, 8, 8, 13, 25, 32, 32, 41, 61, 72, 72, 85, 113, 128, 128, 145, 181, 200, 200, 221, 265, 288, 288, 313, 365, 392, 392, 421, 481, 512, 512, 545, 613, 648, 648, 685, 761, 800, 800, 841, 925, 968, 968, 1013, 1105, 1152, 1152, 1201, 1301, 1352, 1352, 1405, 1513

Offset: 0

Gerhard Kirchner, Jun 28 2020

nonn

P(n) is a corner on a spiral like this:
    * * * * * * * * * * * *
                          *
        * * * * * * * *   *
        *             *   *
        *   * * * *   *   *
        *   *     *   *   *
        *   *   * *   *   *
        *   *         *   *
        *   * * * * * *   *
        *                 *
        * * * * * * * * * *
If we interpret the pointer from P(0) to P(n) as a complex number z(n), the description of the spiral is short because a 90-degree left turn is a multiplication by i (imaginary unit) and the distance of P(n) from P(0) is abs(z(n))^2, see formula 1.

			  n  n*i^(n-1)  z(n)  a(n)
------------------------------------
  0     0        0     0
  1     1        1     1
  2     2i      1+2i   5 = 1^2 + 2^2
  3    -3      -2+2i   8 = 2^2 + 2^2
  4    -4i     -2-2i   8
  5     5       3-2i  13 = 3^2 + 2^2
  6     6i      3+4i  25 = 3^2 + 4^2

Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A000982, A174344, A268038, A274923, A336336.

z[0]=0; z[n_]:=z[n-1]+n*I^(n-1); a[n_]:=z[n]*Conjugate[z[n]]; Array[a,55,0] (* Stefano Spezia, Jun 28 2020 *)

a(n) = abs(z(n))^2 with
  1) z(n) = z(n-1)+n*i^(n-1), z(0)=0.    (recursive)
  2) z(n) = i/2*(n*i^(n+1)-(n+1)*i^n+1). (explicit)
Without complex numbers for k >= 0:
  a(4*k) = 8*k^2,
  a(4*k+1) = 8*k^2+4*k+1,
  a(4*k+2) = 8*k^2+12*k+5,
  a(4*k+3) = 8*(k+1)^2.
From Stefano Spezia, Jun 28 2020: (Start)
G.f.: x*(1 + 2*x - 2*x^2 + 2*x^3 + x^4)/((1 - x)^3*(1 + x^2)^2).
a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n > 6. (End)

A336335 a(n) is the index of the first occurrence of the Euclidean distance prime(n) from a point on a square spiral to its starting point at 1.

Original entry on oeis.org

11, 28, 50, 176, 452, 536, 848, 1388, 2048, 1682, 3752, 4784, 6272, 7268, 8696, 7938, 13748, 14210, 17756, 19952, 11888, 24728, 27308, 25322, 20456, 38888, 42128, 45476, 32792, 49826, 64136, 68252, 43698, 76868, 77930, 90752, 69216, 105788, 111056, 108354, 127628

Offset: 1

Hugo Pfoertner, Jul 24 2020

nonn

			  37--36--35--34--33--32--31
   |                       |
  38  17--16--15--14--13  30  ...
   |   |               |   |   |
  39  18   5---4---3  12  29  54
   |   |   |       |   |   |   |
  40  19   6   1---2 d=2 d=3  53
   |   |   |           |   |   |
  41  20   7---8---9--10  27  52
   |   |                   |   |
  42  21--22--23--24--25--26  51
   |                           |
  43--44--45--46--47--48--49-d=5
.
a(1) = 11 is the index of the first occurrence of distance d = 2 = prime(1) from the start of the spiral.
a(2) = 28 is the index of the first occurrence of distance d = 3 = prime(2) from the start of the spiral.
Distances of the form 4*k+1 corresponding to Pythagorean primes A002144 occur earlier than on the East spoke of the square spiral, dependent on the decomposition of p^2 into two squares. prime(3)^2 = 4^2 + 3^2 leads to index a(3) = 50 in the spiral.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000

Cf. A002144, A002145, A054552, A174344, A268038, A274923, A336336.

a(n) = A054552(prime(n)) if prime(n) != 1 mod 4.

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.

Original entry on oeis.org

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

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A359058 a(n) = squared distance to the origin of the n-th vertex on a counterclockwise undulating spiral in a square grid.

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A345439 Consider the Eisenstein integers x + y*omega, x and y rational integers, represented as the cells of an hexagonal grid; draw a hexagonal spiral as in A345435; a(n) is the norm x^2-x*y+y^2 of the Eisenstein integer in the n-th cell of the spiral.

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PARI

Extensions

A335298 a(n) is the squared distance between the points P(n) and P(0) on a plane, n >= 0, such that the distance between P(n) and P(n+1) is n+1 and, going from P(n) to P(n+2), a 90-degree left turn is taken in P(n+1).

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A336335 a(n) is the index of the first occurrence of the Euclidean distance prime(n) from a point on a square spiral to its starting point at 1.

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Views

Author

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PARI

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Extensions

A345439 Consider the Eisenstein integers x + yomega, x and y rational integers, represented as the cells of an hexagonal grid; draw a hexagonal spiral as in A345435; a(n) is the norm x^2-xy+y^2 of the Eisenstein integer in the n-th cell of the spiral.