cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A228232 Number of strict Gaussian primes of norm less than or equal to n in the first quadrant.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 13, 17, 19, 23, 29, 31, 35, 41, 43, 49, 57, 63, 69, 75, 83, 89, 93, 99, 109, 117, 123, 133, 141, 149, 157, 167, 175, 187, 197, 207, 215, 225, 233, 239, 253, 267, 273, 287, 297, 309, 319, 335, 351, 361, 369, 385, 403, 415, 425, 439, 453, 465, 481, 495
Offset: 1

Views

Author

Olivier Gérard, Aug 17 2013

Keywords

Comments

A Gaussian integer is counted if it has a positive real part and a positive imaginary part (first quadrant excluding the axes).

Links

Crossrefs

Cf. A001182 (number of strict Gaussian integers in the first quadrant).
Cf. A062711 (counts the Gaussian primes on axes also).
Cf. A228233 (version of this sequence including the axes).

Programs

  • Mathematica
    nn = 60; t = Select[Flatten[Table[a + b*I, {a, nn}, {b, nn}]], PrimeQ[#, GaussianIntegers -> True] &]; Table[Length[Select[t, Abs[#] <= n &]], {n, nn}] (* T. D. Noe, Aug 19 2013 *)

A228233 Number of Gaussian primes of norm less than or equal to n in the first quadrant.

Original entry on oeis.org

0, 1, 5, 7, 9, 11, 17, 21, 23, 27, 35, 37, 41, 47, 49, 55, 63, 69, 77, 83, 91, 97, 103, 109, 119, 127, 133, 143, 151, 159, 169, 179, 187, 199, 209, 219, 227, 237, 245, 251, 265, 279, 287, 301, 311, 323, 335, 351, 367, 377, 385, 401, 419, 431, 441, 455, 469
Offset: 1

Views

Author

Olivier Gérard, Aug 17 2013

Keywords

Comments

Include 2 times the primes (once for the real axis, once for the imaginary axis).
More precisely, a(n) includes all Gaussian primes (with the appropriate norms) on the first quadrant's bounding semi-axes. All such Gaussian primes occur in pairs {p, pi} (one real and one imaginary associate), where p is a classical prime of the form 4m + 3 (so p is in A002145) and p <= n. - Rick L. Shepherd, Jun 16 2017

Links

Crossrefs

Cf. A000603 (number of Gaussian integers in the first quadrant with norm less than or equal to n).
Cf. A062711 (counts the Gaussian primes on only one axis).
Cf. A228232 (this sequence excluding classical primes and pure imaginary primes).
Cf. A002145 (Gaussian primes that are positive integers).

Programs

  • Mathematica
    nn = 100; t = Select[Flatten[Table[a + b*I, {a, 0, nn}, {b, 0, nn}]], PrimeQ[#, GaussianIntegers -> True] &]; t2 = Table[0, {nn}]; Do[f = Ceiling[Abs[i]]; If[f <= nn, t2[[f]]++], {i, t}]; Accumulate[t2] (* T. D. Noe, Aug 19 2013 *)

A300416 Number of prime Eisenstein integers z = x - y*w^2 with |z| <= n and where w = -1/2 + i*sqrt(3)/2 is a primitive cube root of unity.

Original entry on oeis.org

0, 2, 4, 6, 9, 11, 15, 17, 23, 25, 30, 34, 40, 44, 50, 54, 61, 65, 71, 79, 87, 91, 98, 104, 114, 122, 128, 138, 147, 155, 161, 171, 183, 193, 199, 209, 217, 225, 237, 249, 262, 276, 286, 296, 308, 318, 331, 345, 359, 365, 377, 391, 410, 418, 428
Offset: 1

Views

Author

Frank M Jackson and Michael B Rees, Mar 05 2018

Keywords

Comments

Two prime Eisenstein integers are not counted separately if they are associated, i.e., if their quotient is a unit (1, -w^2, w, -1, w^2 or -w).

Examples

			a(7)=15 because the Eisenstein primes whose modulus <= 7 are 1-w^2, 1-2w^2, 1-3w^2, 1-5w^2, 1-6w^2, 2, 2-w^2, 2-3w^2, 3-w^2, 3-2w^2, 3-4w^2, 4-3w^2, 5, 5-w^2, 6-w^2.

Links

Crossrefs

Cf. A055667, A062711.

Programs

  • Mathematica
    a[n_] := Module[{w2=-1/2-I*Sqrt[3]/2, lst={}, x, y, z, Nz}, Do[z=x-w2*y; Nz=x^2+x*y+y^2; If[y==0&&Mod[Sqrt[Nz], 3]==2&&Sqrt[Nz]<=n&&PrimeQ[Sqrt[Nz]], AppendTo[lst, {x, y}], If[Mod[Nz, 3]!=2&&Sqrt[Nz]<=n&&PrimeQ[Nz], AppendTo[lst, {x, y}]]], {x, 0, n}, {y, 0, n}]; Length@lst]; Array[a, 100]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.