A055668 -id:A055668 - cp's OEIS frontend

A055664 Norms of Eisenstein-Jacobi primes.

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

N. J. A. Sloane, Jun 09 2000

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

			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.

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.

T. D. Noe, Table of n, a(n) for n = 1..1000
TheGrayCuber, Eisenstein Primes Visually, Youtube video.

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.

Join[{3}, Select[Range[600], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) & ]] (* Jean-François Alcover, Oct 09 2012, from formula *)

is(n)=(isprime(n) && n%3<2) || (issquare(n,&n) && isprime(n) && n%3==2) \\ Charles R Greathouse IV, Apr 30 2013

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

More terms from David Wasserman, Mar 21 2002

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

N. J. A. Sloane, Jun 09 2000

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).

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

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.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Gaussian integers and primes

Cf. A055025, A055026, A055027, A055028.

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

a055029 2 = 1
a055029 n = 2 * a079260 n + a079261 (a037213 n)
-- Reinhard Zumkeller, Nov 11 2012

a[n_ /; PrimeQ[n] && Mod[n, 4] == 1] = 2; a[2] = 1; a[n_ /; (p = Sqrt[n]; PrimeQ[p] && Mod[p, 4] == 3)] = 1; a[] = 0; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover, Oct 25 2011, after Franklin T. Adams-Watters *)

a(n)=if(isprime(n), if(n%4==1, 2, n==2), if(issquare(n, &n) && isprime(n) && n%4==3, 1, 0)) \\ Charles R Greathouse IV, Feb 07 2017

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

More terms from Reiner Martin, Jul 20 2001

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

N. J. A. Sloane, Jun 09 2000

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

			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.

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.

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

a[3] = 6; a[p_ /; PrimeQ[p] && Mod[p, 6] == 1] = 12; a[n_ /; PrimeQ[p = Sqrt[n]] && Mod[p, 3] == 2] = 6; a[] = 0; Table[a[n], {n, 0, 99}] (* _Jean-François Alcover, Oct 24 2013, after Franklin T. Adams-Watters *)

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

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

A055666 Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

N. J. A. Sloane, Jun 09 2000

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

Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-omega, +-omega^2).

			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.

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.

Jean-François Alcover, Table of n, a(n) for n = 1..1000

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

norms = Join[{3}, Select[Range[2000], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) &]]; r[n_] := Length[Reduce[n == a^2 - a*b + b^2, {a, b}, Integers]]/6; A055666 = r /@ norms (* Jean-François Alcover, Oct 24 2013 *)

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

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

N. J. A. Sloane, Jun 09 2000

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

			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.

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.

Antti Karttunen, Table of n, a(n) for n = 1..1000 (computed from the b-file of A055666 with the formula of _Franklin T. Adams-Watters_)

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

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 *)

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

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

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

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

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

Data corrected and name clarified by Rémy Sigrist, Aug 08 2018

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A055664 Norms of Eisenstein-Jacobi primes.

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A055029 Number of inequivalent Gaussian primes of norm n.

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A055667 Number of Eisenstein-Jacobi primes of norm n.

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A055666 Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).

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

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

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