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 1-4 of 4 results.

A055025 Norms of Gaussian primes.

Original entry on oeis.org

2, 5, 9, 13, 17, 29, 37, 41, 49, 53, 61, 73, 89, 97, 101, 109, 113, 121, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 361, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 529, 541, 557, 569
Offset: 1

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Author

N. J. A. Sloane, Jun 09 2000

Keywords

Comments

This is the range of the norm a^2 + b^2 of Gaussian primes a + b i. A239621 lists for each norm value a(n) one of the Gaussian primes as a, b with a >= b >= 0. In A239397, any of these (a, b) is followed by (b, a), except for a = b = 1. - Wolfdieter Lang, Mar 24 2014, edited by M. F. Hasler, Mar 09 2018
From Jean-Christophe Hervé, May 01 2013: (Start)
The present sequence is related to the square lattice, and to its division in square sublattices. Let's say that an integer n divides a lattice if there exists a sublattice of index n. Example: 2, 4, 5 divide the square lattice. Then A001481 (norms of Gaussian integers) is the sequence of divisors of the square lattice. Say that n is a "prime divisor" if the index-n sublattice is not contained in any other sublattice except the original lattice itself. The present sequence gives the "prime divisors" of the square lattice.
Similarly, A055664 (Norms of Eisenstein-Jacobi primes) is the sequence of "prime divisors" of the hexagonal lattice. (End)
The sequence is formed of 2, the prime numbers of form 4k+1, and the square of other primes (of form 4k+3). These are the primitive elements of A001481. With 0 and 1, they are the numbers that are uniquely decomposable in the sum of two squares. - Jean-Christophe Hervé, Nov 17 2013

Examples

			There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i). In A239621 2+i is listed as 2, 1.

References

  • R. K. Guy, Unsolved Problems in Number Theory, A16.
  • L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.

Links

Crossrefs

Cf. A055026, A055027, A055028, A055029, A055664-A055666, A001481.
Cf. A239397, A239621 (Gaussian primes).

Programs

  • Mathematica
    Union[(#*Conjugate[#] & )[ Select[Flatten[Table[a + b*I, {a, 0, 23}, {b, 0, 23}]], PrimeQ[#, GaussianIntegers -> True] & ]]][[1 ;; 55]] (* Jean-François Alcover, Apr 08 2011 *)
(* Or, from formula: *) maxNorm = 569; s1 = Select[Range[1, maxNorm, 4], PrimeQ]; s3 = Select[Range[3, Sqrt[maxNorm], 4], PrimeQ]^2; Union[{2}, s1, s3]  (* Jean-François Alcover, Dec 07 2012 *)
  • PARI
    list(lim)=my(v=List()); if(lim>=2, listput(v,2)); forprime(p=3,sqrtint(lim\1), if(p%4==3, listput(v,p^2))); forprime(p=5,lim, if(p%4==1, listput(v,p))); Set(v) \\ Charles R Greathouse IV, Feb 06 2017
  • PARI
    isA055025(n)=(isprime(n) && n%4<3) || (issquare(n, &n) && isprime(n) && n%4==3) \\ Jianing Song, Aug 15 2023, based on Charles R Greathouse IV's program for A055664

Formula

Consists of 2; rational primes = 1 (mod 4) [A002144]; and squares of rational primes = 3 (mod 4) [A002145^2].
a(n) ~ 2n log n. - Charles R Greathouse IV, Feb 06 2017

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

A055029 Number of inequivalent Gaussian primes of norm n.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane, Jun 09 2000

Keywords

Comments

These are the primes in the ring of integers a+bi, a and b rational integers, i = sqrt(-1).
Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-i).

Examples

			There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i).

References

  • R. K. Guy, Unsolved Problems in Number Theory, A16.
  • L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.

Links

Crossrefs

Cf. A055025, A055026, A055027, A055028.
Cf. A055664, A055665, A055666, A055667, A055668.

Programs

Formula

a(n) = A055028(n)/4.
a(n) = 2 if n is a prime = 1 (mod 4); a(n) = 1 if n is 2, or p^2 where p is a prime = 3 (mod 4); a(n) = 0 otherwise. - Franklin T. Adams-Watters, May 05 2006
a(n) = if n = 2 then 1 else 2*A079260(n) + A079261(A037213(n)). - Reinhard Zumkeller, Nov 11 2012

Extensions

More terms from Reiner Martin, Jul 20 2001

A055665 Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Original entry on oeis.org

6, 6, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12
Offset: 1

Views

Author

N. J. A. Sloane, Jun 09 2000

Keywords

Comments

These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.

Examples

			There are 6 Eisenstein-Jacobi primes of norm 3, omega-omega^2 times one of the 6 units [ +-1, +-omega, +-omega^2 ] but only one up to equivalence.

References

  • R. K. Guy, Unsolved Problems in Number Theory, A16.
  • L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.

Links

Crossrefs

Cf. A055664-A055668, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

Programs

  • Mathematica
    norms = Join[{3}, Select[Range[1000], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) &]]; r[n_] := Reduce[n == a^2 - a*b + b^2, {a, b}, Integers] // Length; A055665 = r /@ norms (* Jean-François Alcover, Oct 24 2013 *)

Formula

a(n) = 6 * A055666(n) - Franklin T. Adams-Watters, May 05 2006

Extensions

More terms from Franklin T. Adams-Watters, May 05 2006

A134324 Number of Eisenstein-Jacobi primes whose modulus is > n and <= n+1.

Original entry on oeis.org

0, 12, 12, 12, 18, 12, 24, 12, 36, 12, 30, 24, 36, 24, 36, 24, 42, 24, 36, 48, 48, 24, 42, 36, 60, 48, 36, 60, 54, 48, 36, 60, 72, 60, 36, 60, 48, 48, 72, 72, 78, 84, 60, 60, 72, 60, 78, 84, 84, 36, 72, 84, 114, 48
Offset: 0

Views

Author

Philippe Lallouet (philip.lallouet(AT)orange.fr), Jan 30 2008, Feb 06 2008

Keywords

Crossrefs

Cf. A055664, A055665, A055666, A055667, A055668.

Formula

a(n) = Sum_{k=n^2+1..(n+1)^2} A055667(k). - Rémy Sigrist, Aug 08 2018

Extensions

Data corrected and name clarified by Rémy Sigrist, Aug 08 2018
Showing 1-4 of 4 results.

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

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