A055667 -id:A055667 - cp's OEIS frontend

A055029 Number of inequivalent Gaussian primes of norm n.

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

A345435 Represent the ring of Eisenstein integers E = {x+yomega: x, y rational integers, omega = exp(2Pi*i/3)} by the cells of a hexagonal grid; number the cells of the grid along a counterclockwise hexagonal spiral, with the cells 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

Original entry on oeis.org

0, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 90, 91, 95, 97, 101, 103, 107, 109, 113, 115

Offset: 1

N. J. A. Sloane, Jun 23 2021

nonn

The Eisenstein integer represented by cell m is A307013(m) + A307012(m)*omega. Thus the set of Eisenstein primes is {A307013(a(n)) + A307012(a(n))*omega : n >= 2}. - Peter Munn, Jun 26 2021
The Eisenstein integer a + b*omega has norm a^2 - a*b + b^2 (see A003136). The number of Eisenstein integers of norm n is given by A004016(n).
The norms of the Eisenstein primes are given in A055664, and the number of Eisenstein primes of norm n is given in A055667.
Reid's 1910 book (still in print) is still the best reference for the Eisenstein integers and similar rings.

			The smallest Eisenstein integers are 0 (of norm 0), and the six units of norm 1, namely (writing w for omega) +-1, +-w, +-w^2.
The first few Eisenstein primes are (here u is any of the six units):
   u*(2+w), norm = 3, number = 6;
   2*u, norm = 4, number = 6;
   u*(3+w), norm = 7, number = 6;
   u*(3+2*w), norm = 7, number = 6 (so there are 12 primes of norm 7 - see A055667).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.4, p. 111.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A345435
N. J. A. Sloane, Illustration of initial terms [An enlargement of Figure 1 of Wichmann (2019), showing the numbering of the initial cells of the hexagonal spiral.]
Eric Weisstein's World of Mathematics, Eisenstein Prime
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019.
Brian Wichmann, The Eisenstein integers, with the primes shaded [Figure 1 from the previous link]

Cf. A003136, A003173, A003627, A004016, A007645, A055664, A055667, A307012, A307013, A308412, A345436, A345437.

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More terms from Rémy Sigrist, Jun 26 2021

A055668 Number of inequivalent Eisenstein-Jacobi primes of norm n.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 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.

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.

Cf. A055664-A055667, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

a[3] = 1; a[p_ /; PrimeQ[p] && Mod[p, 6] == 1] = 2; a[n_ /; PrimeQ[p = Sqrt[n]] && Mod[p, 3] == 2] = 1; a[] = 0; Table[a[n], {n, 0, 104}] (* _Jean-François Alcover, Aug 19 2013, after Franklin T. Adams-Watters *)
Table[Which[PrimeQ[n]&&Mod[n,6]==1,2,n==3,1,PrimeQ[Sqrt[n]]&&Mod[ Sqrt[ n],3] == 2,1,True,0],{n,0,110}] (* Harvey P. Dale, Jun 17 2017 *)

a(n) = 2 if n is a prime = 1 (mod 6); a(n) = 1 if n = 3 or n = p^2 where p is a prime = 2 (mod 3); a(n) = 0 otherwise. - Franklin T. Adams-Watters, May 05 2006

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

A135461 a(n) = 1 if n is the norm of an Eisenstein prime (see A055664) otherwise 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

N. J. A. Sloane, Feb 06 2008

nonn

			The smallest primes are 1-omega (of norm 3) and 2 (of norm 4).

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 = 0..65537
Index entries for characteristic functions

Cf. A055667, A135462.

Characteristic function of A055664.

f[n_]:=If[ PrimeQ[n] && Mod[n, 6] == 1|| PrimeQ[Sqrt[n]] && Mod[Sqrt[n], 3] == 2||n==3,1,0];Array[f,99,0] (* James C. McMahon, Apr 15 2025 *)

A135461(n) = (isprime(n) && n%3<2) || (issquare(n, &n) && isprime(n) && n%3==2); \\ This is Charles R Greathouse IV's Apr 30 2013 code (with name "is") for A055664. - Antti Karttunen, Dec 06 2017

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

N. J. A. Sloane, Feb 06 2008

nonn

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

Cf. A135461, A055664, A055667.

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 *)

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

Terms corrected by Seiichi Manyama, Dec 09 2017

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

A300416 Number of prime Eisenstein integers z = x - yw^2 with |z| <= n and where w = -1/2 + isqrt(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

Frank M Jackson and Michael B Rees, Mar 05 2018

nonn

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).

			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.

Eric Weisstein's World of Mathematics, Eisenstein prime
Wikipedia, Eisenstein integer.

Cf. A055667, A062711.

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]

cp's OEIS Frontend

A055029 Number of inequivalent Gaussian primes of norm n.

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A055668 Number of inequivalent Eisenstein-Jacobi primes of norm n.

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A135461 a(n) = 1 if n is the norm of an Eisenstein prime (see A055664) otherwise 0.

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

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

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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.

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