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.

A055664 Norms of Eisenstein-Jacobi primes.

Original entry on oeis.org

3, 4, 7, 13, 19, 25, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 121, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 289, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 529, 541, 547, 571
Offset: 1

Views

Author

N. J. A. Sloane, Jun 09 2000

Keywords

Comments

These are the norms of the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.
Let us say that an integer n divides a lattice if there exists a sublattice of index n. Example: 3 divides the hexagonal lattice. Then A003136 (Loeschian numbers) is the sequence of divisors of the hexagonal 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 hexagonal lattice. Similarly, A055025 (Norms of Gaussian primes) is the sequence of "prime divisors" of the square lattice. - Jean-Christophe Hervé, Dec 04 2006

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. A055665-A055668, A055025-A055029, A135461, A135462. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.
The Z[sqrt(-5)] analogs are in A020669, A091727, A091728, A091729, A091730 and A091731.

Programs

  • Mathematica
    Join[{3}, Select[Range[600], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) & ]] (* Jean-François Alcover, Oct 09 2012, from formula *)
  • PARI
    is(n)=(isprime(n) && n%3<2) || (issquare(n,&n) && isprime(n) && n%3==2) \\ Charles R Greathouse IV, Apr 30 2013

Formula

Consists of 3; rational primes == 1 (mod 3) [A002476]; and squares of rational primes == -1 (mod 3) [A003627^2].

Extensions

More terms from David Wasserman, Mar 21 2002

A055667 Number of Eisenstein-Jacobi primes of norm n.

Original entry on oeis.org

0, 0, 0, 6, 6, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 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+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.

Crossrefs

Cf. A055664-A055668, A055025-A055029, A135461, A135462. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

Programs

Formula

a(n) = 6 * A055668(n). - Franklin T. Adams-Watters, May 05 2006

Extensions

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

A135462 a(n) = number of Eisenstein primes (see A055664) of norm <= n.

Original entry on oeis.org

0, 0, 0, 6, 12, 12, 12, 24, 24, 24, 24, 24, 24, 36, 36, 36, 36, 36, 36, 48, 48, 48, 48, 48, 48, 54, 54, 54, 54, 54, 54, 66, 66, 66, 66, 66, 66, 78, 78, 78, 78, 78, 78, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 102, 102, 102, 102, 102
Offset: 0

Views

Author

N. J. A. Sloane, Feb 06 2008

Keywords

References

  • L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.

Crossrefs

Cf. A135461, A055664, A055667.

Programs

  • Mathematica
    A055667[n_] := Which[n == 3, 6, PrimeQ[n] && Mod[n, 6] == 1, 12, PrimeQ[p = Sqrt[n]] && Mod[p, 3] == 2, 6, True, 0]; Accumulate[Array[A055667, 66, 0]] (* Jean-François Alcover, Feb 28 2018 *)

Formula

a(n) = a(n-1) + A055667(n) for n > 0. - Seiichi Manyama, Dec 09 2017

Extensions

Terms corrected by Seiichi Manyama, Dec 09 2017
Showing 1-3 of 3 results.

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

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