A319446 Exponent of the group of the Eisenstein integers in a reduced system modulo n.

1, 3, 6, 6, 24, 6, 6, 12, 6, 24, 120, 6, 12, 6, 24, 24, 288, 6, 18, 24, 6, 120, 528, 12, 120, 12, 18, 6, 840, 24, 30, 48, 120, 288, 24, 6, 36, 18, 12, 24, 1680, 6, 42, 120, 24, 528, 2208, 24, 42, 120, 288, 12, 2808, 18, 120, 12, 18, 840, 3480, 24, 60

Offset

1,2

Comments

Equivalent of psi (A002322) in the ring of Eisenstein integers.

a(n) is the exponent of the multiplicative group of Eisenstein integers modulo n, i.e., (Z[w]/nZ[w])* = {a + b*w : a, b in Z/nZ and gcd(a^2 + a*b + b^2, n) = 1} where w = (1 + sqrt(3)*i)/2. The number of elements in (Z[w]/nZ[w])* is A319445(n).

a(n) is the smallest e such that for any Eisenstein integer z coprime to n we have z^e == 1 (mod n).

By definition, A319445(n)/a(n) is always an integer, and is 1 iff (Z[w]/nZ[w])* is cyclic, that is, rank((Z[w]/nZ[w])*) = A319447(n) = 0 or 1, and n has a primitive root in (Z[w]/nZ[w])*. A319445(n)/a(n) = 1 iff n = 1, 3 or a prime congruent to 2 mod 3.

For n > 2, a(n) is divisible by 6.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Wikipedia, Eisenstein integer

Wikipedia, Torsion group

Formula

a(3) = 6, a(3^e) = 2*3^(e-1) for e >= 2; a(p^e) = (p - 1)*p^(e-1) if p == 1 (mod 3) and (p^2 - 1)*p^(e-1) if p == 2 (mod 3). If gcd(m, n) = 1 then a(mn) = lcm(a(m), a(n)). [See the group structure of (Z[w]/(pi^e)Z[w])* in A319447, where pi is a prime element in Z[w]. - Jianing Song, Oct 03 2022]

Example

Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Let G = (Z[w]/4Z[w])* = {1, w, 1 + w, w', 1 + w', -1 + 2w, -1, -w, -1 - w, -w', -1 - w', -1 + 2w'}. The possibilities for the exponent of G are 12, 6, 4, 3, 2 and 1. G^6 = {x^6 mod 4 : x belongs to G} = {1} and w^3 !== 1 (mod 4), w^4 !== 1 (mod 4). Therefore, the exponent of G is greater than 4, accordingly the exponent of G is 6 and a(4) = 6.

Mathematica

f[p_, e_] := If[p == 3 , If[e == 1, 6, 2*3^(e - 1)], Switch[Mod[p, 3], 1, (p - 1)*p^(e - 1), 2, (p^2 - 1)*p^(e - 1)]]; eisPsi[1] = 1; eisPsi[n_] := LCM @@ f @@@ FactorInteger[n]; Array[eisPsi, 100]  (* Amiram Eldar, Feb 10 2020 *)

Prog

(PARI)
a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==3, r=lcm(r, 2*3^max(e-1, 1)));
        if(p%3==1, r=lcm(r, (p-1)*p^(e-1)));
        if(p%3==2, r=lcm(r, (p^2-1)*p^(e-1)));
    );
    return(r);
}

Crossrefs

Equivalent of arithmetic functions in the ring of Eisenstein integers (the corresponding functions in the ring of integers are in the parentheses): A319442 ("d", A000005), A319449 ("sigma", A000203), A319445 ("phi", A000010), this sequence ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).

Equivalent in the ring of Gaussian integers: A227334.

Sequence in context: A265026 A304335 A223048 * A181372 A168426 A065931

Adjacent sequences: A319443 A319444 A319445 * A319447 A319448 A319449

Keyword

nonn

Author

Jianing Song, Sep 19 2018

Extensions

Corrected by Jianing Song, Jan 12 2019

Status

approved