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A319446 Exponent of the group of the Eisenstein integers in a reduced system modulo n. 8
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 (list; graph; refs; listen; history; text; internal format)
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)).

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

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Last modified July 6 23:01 EDT 2020. Contains 335484 sequences. (Running on oeis4.)