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%I #33 Oct 03 2022 15:08:17
%S 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,
%T 12,18,6,840,24,30,48,120,288,24,6,36,18,12,24,1680,6,42,120,24,528,
%U 2208,24,42,120,288,12,2808,18,120,12,18,840,3480,24,60
%N Exponent of the group of the Eisenstein integers in a reduced system modulo n.
%C Equivalent of psi (A002322) in the ring of Eisenstein integers.
%C 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).
%C a(n) is the smallest e such that for any Eisenstein integer z coprime to n we have z^e == 1 (mod n).
%C 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.
%C For n > 2, a(n) is divisible by 6.
%H Jianing Song, <a href="/A319446/b319446.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Torsion_group">Torsion group</a>
%F 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]
%e Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
%e 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.
%t 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 *)
%o (PARI)
%o a(n)=
%o {
%o my(r=1, f=factor(n));
%o for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
%o if(p==3, r=lcm(r,2*3^max(e-1,1)));
%o if(p%3==1, r=lcm(r,(p-1)*p^(e-1)));
%o if(p%3==2, r=lcm(r,(p^2-1)*p^(e-1)));
%o );
%o return(r);
%o }
%Y 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).
%Y Equivalent in the ring of Gaussian integers: A227334.
%K nonn
%O 1,2
%A _Jianing Song_, Sep 19 2018
%E Corrected by _Jianing Song_, Jan 12 2019
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