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.

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 *)

A228234 Number of strict Gaussian primes of norm less than or equal to n in the first quadrant on or below the first diagonal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 9, 10, 12, 15, 16, 18, 21, 22, 25, 29, 32, 35, 38, 42, 45, 47, 50, 55, 59, 62, 67, 71, 75, 79, 84, 88, 94, 99, 104, 108, 113, 117, 120, 127, 134, 137, 144, 149, 155, 160, 168, 176, 181, 185, 193, 202, 208, 213, 220, 227, 233, 241, 248, 256
Offset: 1

Views

Author

Olivier Gérard, Aug 17 2013

Keywords

Comments

Strict means that one does not include the ordinary integer primes and integer primes multiplied by i.
In the first quadrant and on or below the first diagonal, means here that the imaginary part is positive and inferior or equal to the real part.

Links

Crossrefs

Cf. A211340 (number of strict Gaussian integers in this half-quadrant).
Cf. A228235 (a version of this sequence including the real axis).
Cf. A228232, A228233 (versions counting the whole first quadrant).

Programs

  • Mathematica
    nn = 100; t = Select[Flatten[Table[a + b*I, {a, nn}, {b, a, 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 *)

A228235 Number of Gaussian primes of norm less than or equal to n in the first quadrant on or below the first diagonal.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 9, 11, 12, 14, 18, 19, 21, 24, 25, 28, 32, 35, 39, 42, 46, 49, 52, 55, 60, 64, 67, 72, 76, 80, 85, 90, 94, 100, 105, 110, 114, 119, 123, 126, 133, 140, 144, 151, 156, 162, 168, 176, 184, 189, 193, 201, 210, 216, 221, 228, 235, 241, 250, 257
Offset: 1

Views

Author

Olivier Gérard, Aug 17 2013

Keywords

Comments

In the first quadrant and on or below the first diagonal, means here that the imaginary part is nonnegative and inferior or equal to the real part.
The norm used is the absolute value of the Gaussian integers, seen as complex numbers : sqrt( re(z)^2 + im(z)^2).

Links

Crossrefs

Cf. A228172 (number of Gaussian integers in this half-quadrant).
Cf. A228234 (version of this sequence excluding the real axis).
Cf. A228232, A228233 (versions counting the whole first quadrant).

Programs

  • Mathematica
    nn = 100; t = Select[Flatten[Table[a + b*I, {a, 0, nn}, {b, a, 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 *)
Showing 1-3 of 3 results.

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

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