A055029 -id:A055029 - cp's OEIS frontend

A118916 Number of inequivalent primes in ring of integers Z[sqrt(2)] of successive norms (indexed by A055029).

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

Offset: 1

Franklin T. Adams-Watters, May 05 2006

Cf. A055029, A118917.

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

N. J. A. Sloane, Jun 09 2000

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

			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.

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Gaussian prime
Wikipedia, Table of Gaussian integer factorizations
Index entries for Gaussian integers and primes

Cf. A055026, A055027, A055028, A055029, A055664-A055666, A001481.

Cf. A239397, A239621 (Gaussian primes).

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 *)

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

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

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

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

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

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

A055673 Absolute values of norms of primes in ring of integers Z[sqrt(2)].

Original entry on oeis.org

2, 7, 9, 17, 23, 25, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 121, 127, 137, 151, 167, 169, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 361, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593

Offset: 1

N. J. A. Sloane, Jun 09 2000

The integers have the form z = a + b*sqrt(2), a and b rational integers. The norm of z is a^2 - 2*b^2, which may be negative.

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

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A055026-A055029, A055669-A055672, A118916, A118917.

maxNorm = 593; s1 = Select[Range[-1, maxNorm, 8], PrimeQ]; s2 = Select[Range[1, maxNorm, 8], PrimeQ]; s3 = Select[Range[-3, Sqrt[maxNorm], 8], PrimeQ]^2; s4 = Select[Range[3, Sqrt[maxNorm], 8], PrimeQ]^2; Union[{2}, s1, s2, s3, s4] (* Jean-François Alcover, Dec 07 2012, from formula *)

is(n)=!!if(isprime(n), setsearch([1,2,7],n%8), issquare(n,&n) && isprime(n) && setsearch([3,5], n%8)) \\ Charles R Greathouse IV, Sep 10 2016

Consists of 2; rational primes = +-1 (mod 8); and squares of rational primes = +-3 (mod 8).

I would also like to get the sequences (analogous to A055027 and A055029) giving the number of inequivalent primes mod units. Of course now there are infinitely many units.

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

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

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

A055026 Number of Gaussian primes of successive norms (indexed by A055025).

Original entry on oeis.org

4, 8, 4, 8, 8, 8, 8, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 8, 8

Offset: 1

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).

			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.

Index entries for Gaussian integers and primes

Cf. A055025-A055029, A055664-...

m = 32; Length /@ Split[Sort[Select[Flatten[Table[{a^2 + b^2, a + b*I}, {a, -m, m}, {b, -m, m}], 1], PrimeQ[#[[2]], GaussianIntegers -> True] & ]], #1[[1]] == #2[[1]] & ][[1 ;; 87]] (* Jean-François Alcover, Apr 08 2011 *)

More terms from Reiner Martin, Jul 20 2001

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

A295996 One quarter of number of Gaussian primes whose norm is 4*n+1 or less.

Original entry on oeis.org

0, 3, 4, 6, 8, 8, 8, 10, 10, 12, 14, 14, 15, 17, 17, 19, 19, 19, 21, 21, 21, 21, 23, 23, 25, 27, 27, 29, 31, 31, 32, 32, 32, 32, 34, 34, 34, 36, 36, 38, 38, 38, 38, 40, 40, 42, 42, 42, 44, 46, 46, 46, 46, 46, 46, 46, 46, 48, 50, 50, 52, 52, 52, 52, 54, 54, 54, 56

Offset: 0

Seiichi Manyama, Dec 02 2017

nonn

			The Gaussian primes whose norm is 9 or less;
      *        3i,
    *   *      -1+2i, 1+2i
  * *   * *    -2+i, -1+i, 1+i, 2+i
*           *  -3, 3
  * *   * *    -2-i, -1-i, 1-i, 2-i
    *   *      -1-2i, 1-2i
      *        -3i
               a(2) = 16/4 = 4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Gaussian Prime

Cf. A016813, A055029, A091100, A091134, A135462, A296020, A296021.

require 'prime'
def A(k, n)
  ary = []
  cnt = 0
  k.step(4 * n + k, 4){|i|
    cnt += 1 if i.prime?
    ary << cnt
  }
  ary
end
def A295996(n)
  ary1 = A(1, n)
  ary3 = A(3, Math.sqrt(n).to_i) + [0]
  [0] + (1..n).map{|i| 1 + 2 * ary1[i] + ary3[(Math.sqrt(4 * i + 1).to_i - 3) / 4]}
end
p A295996(100)

A055027 Number of inequivalent Gaussian primes of successive norms (indexed by A055025).

Original entry on oeis.org

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

Offset: 1

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.

Index entries for Gaussian integers and primes

Cf. A055025-A055029, A055664-...

norms = Union[ #*Conjugate[#]& [ Select[ Flatten[ Table[a + b*I, {a, 0, 31}, {b, 0, 31}]], PrimeQ[#, GaussianIntegers -> True] &]]]; f[norm_] := (Clear[a, b]; primes = {a + b*I} /. {ToRules[ Reduce[a^2 + b^2 == norm, {a, b}, Integers]]}; primes //. {p1___, p2_, p3___, p4_, p5___} /; MatchQ[p2, (-p4 | I*p4 | -I*p4)] :> {p1, p2, p3, p5} // Length); A055027 = f /@ norms (* Jean-François Alcover, Nov 30 2012 *)

More terms from Reiner Martin, Jul 20 2001

cp's OEIS Frontend

A118916 Number of inequivalent primes in ring of integers Z[sqrt(2)] of successive norms (indexed by A055029).

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A055025 Norms of Gaussian primes.

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A055664 Norms of Eisenstein-Jacobi primes.

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A055673 Absolute values of norms of primes in ring of integers Z[sqrt(2)].

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A055667 Number of Eisenstein-Jacobi primes of norm n.

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

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A055026 Number of Gaussian primes of successive norms (indexed by A055025).

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