(Translated by https://www.hiragana.jp/)
A055668 -id:A055668 - OEIS
login
The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a055668 -id:a055668
Displaying 1-6 of 6 results found.
page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A055664
Norms of Eisenstein-Jacobi primes.
+10
27
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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
T. D. Noe, Table of n, a(n) for n = 1..1000
FORMULA
Consists of 3; rational primes == 1 (mod 3) [A002476]; and squares of rational primes == -1 (mod 3) [A003627^2].
EXAMPLE
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.
MATHEMATICA
Join[{3}, Select[Range[600], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) & ]] (* Jean-François Alcover, Oct 09 2012, from formula *)
PROG
(PARI) is(n)=(isprime(n) && n%3<2) || (issquare(n, &n) && isprime(n) && n%3==2) \\ Charles R Greathouse IV, Apr 30 2013
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.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
More terms from David Wasserman, Mar 21 2002
STATUS
approved
A055029
Number of inequivalent Gaussian primes of norm n.
+10
14
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
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).
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
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Gaussian integers and primes
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
EXAMPLE
There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i).
MATHEMATICA
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 *)
PROG
(Haskell)
a055029 2 = 1
a055029 n = 2 * a079260 n + a079261 (a037213 n)
-- Reinhard Zumkeller, Nov 11 2012
(PARI) 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
CROSSREFS
Cf. A055025, A055026, A055027, A055028.
Cf. A055664, A055665, A055666, A055667, A055668.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
More terms from Reiner Martin, Jul 20 2001
STATUS
approved
A055667
Number of Eisenstein-Jacobi primes of norm n.
+10
7
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.
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
Table of n, a(n) for n=0..99.
FORMULA
a(n) = 6 * A055668(n). - Franklin T. Adams-Watters, May 05 2006
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
Cf. A055664-A055668, A055025-A055029, A135461, A135462. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
More terms from Franklin T. Adams-Watters, May 05 2006
STATUS
approved
A055666
Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).
+10
5
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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).
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
Jean-François Alcover, Table of n, a(n) for n = 1..1000
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
Cf. A055664-A055668, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
More terms from Franklin T. Adams-Watters, May 05 2006
STATUS
approved
A055665
Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).
+10
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.
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
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)
FORMULA
a(n) = 6 * A055666(n) - Franklin T. Adams-Watters, May 05 2006
EXAMPLE
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.
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 *)
CROSSREFS
Cf. A055664-A055668, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
More terms from Franklin T. Adams-Watters, May 05 2006
STATUS
approved
A134324
Number of Eisenstein-Jacobi primes whose modulus is > n and <= n+1.
+10
0
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..53.
FORMULA
a(n) = Sum_{k=n^2+1..(n+1)^2} A055667(k). - Rémy Sigrist, Aug 08 2018
CROSSREFS
Cf. A055664, A055665, A055666, A055667, A055668.
KEYWORD
easy,nonn
AUTHOR
Philippe Lallouet (philip.lallouet(AT)orange.fr), Jan 30 2008, Feb 06 2008
EXTENSIONS
Data corrected and name clarified by Rémy Sigrist, Aug 08 2018
STATUS
approved
page 1

Search completed in 0.009 seconds

Lookup Welcome Wiki Register Music Plot 2 Demos Index WebCam Contribute Format Style Sheet Transforms Superseeker Recents
The OEIS Community
Maintained by The OEIS Foundation Inc.
Last modified October 10 08:54 EDT 2024. Contains 376632 sequences.
License Agreements, Terms of Use, Privacy Policy