A227334 Exponent of the group of the Gaussian integers in a reduced system modulo n.

1, 2, 8, 4, 4, 8, 48, 4, 24, 4, 120, 8, 12, 48, 8, 8, 16, 24, 360, 4, 48, 120, 528, 8, 20, 12, 72, 48, 28, 8, 960, 16, 120, 16, 48, 24, 36, 360, 24, 4, 40, 48, 1848, 120, 24, 528, 2208, 8, 336, 20, 16, 12, 52, 72, 120, 48, 360, 28, 3480, 8, 60, 960, 48, 32

Offset

1,2

Comments

a(n) is the exponent of the multiplicative group of Gaussian integers modulo n, i.e., (Z[i]/nZ[i])* = {a + b*i: a, b in Z/nZ and gcd(a^2 + b^2, n) = 1}. The number of elements in (Z[i]/nZ[i])* is A079458(n).

For n > 2, a(n) is divisible by 4. - Jianing Song, Aug 29 2018

From Jianing Song, Sep 23 2018: (Start)

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

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

By definition, A079458(n)/a(n) is always an integer, and is 1 iff (Z[i]/nZ[i])* is cyclic, that is, rank((Z[i]/nZ[i])*) = A316506(n) = 0 or 1, and n has a primitive root in (Z[i]/nZ[i])*. A079458(n)/a(n) = 1 iff n = 1, 2 or a prime congruent to 3 modulo 4. (End)

Links

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

Wikipedia, Gaussian integer

Wikipedia, Torsion group

Formula

a(2^e) = 2^e if e <= 2 and 2^(e-1) if e >= 3, a(p^e) = (p - 1)*p^(e-1) if p == 1 (mod 4) and (p^2 - 1)*p^(e-1) if p == 3 (mod 4). If gcd(m, n) = 1 then a(mn) = lcm(a(m), a(n)). - Jianing Song, Aug 29 2018 [See the group structure of (Z[i]/(pi^e)Z[i])* in A316506, where pi is a prime element in Z[i]. - Jianing Song, Oct 03 2022]

Example

Let G = (Z[i]/4Z[i])* = {i, 3i, 1, 1 + 2i, 2 + i, 2 + 3i, 3, 3 + 2i}. The possibilities for the exponent of G are 8, 4, 2 and 1. G^4 = {x^4 mod 4 : x belongs to G} = {1} and i^2 !== 1 (mod 4). Therefore, the exponent of G is greater than 2, accordingly the exponent of G is 4 and a(4) = 4.

Mathematica

fa = FactorInteger; lamas[1] = 1; lamas[p_, s_]:= Which[Mod[p, 4]==3, p^(s-1)(p^2 - 1), Mod[p, 4] == 1, p^(s - 1)(p - 1), s ≥ 4, 2^(s - 1), s > 1, 4, s == 1, 2]; lamas[n_] := {aux = 1; Do[aux = LCM[aux, lamas[fa[n][[i, 1]], fa[n][[i, 2]]]], {i, 1, Length@fa[n]}]; aux}[[1]]; Table[lamas[n], {n, 100}]

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==2&&e<=2, r=lcm(r, 2^e));
        if(p==2&&e>=3, r=lcm(r, 2^(e-1)));
        if(p%4==1, r=lcm(r, (p-1)*p^(e-1)));
        if(p%4==3, r=lcm(r, (p^2-1)*p^(e-1)));
    );
    return(r);
} \\ Jianing Song, Aug 29 2018

Crossrefs

Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), this sequence ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).

Equivalent in the ring of Eisenstein integers: A319446.

Sequence in context: A086311 A194940 A195920 * A105351 A178592 A348683

Adjacent sequences: A227331 A227332 A227333 * A227335 A227336 A227337

Keyword

nonn

Author

José María Grau Ribas, Jul 07 2013

Status

approved