

A055664


Norms of EisensteinJacobi primes.


16



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
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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's 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 indexn 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.  JeanChristophe 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 EisensteinJacobi primes of norm 3, omegaomega^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) & ]] (* JeanFrançois 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. A055665A055668, A055025A055029, 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.
Sequence in context: A279815 A088764 A093124 * A089374 A029552 A193883
Adjacent sequences: A055661 A055662 A055663 * A055665 A055666 A055667


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane, Jun 09 2000


EXTENSIONS

More terms from David Wasserman, Mar 21 2002


STATUS

approved



