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Number of Eisenstein integers in a reduced system modulo n.
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%I #26 Feb 13 2024 02:20:23

%S 1,3,6,12,24,18,36,48,54,72,120,72,144,108,144,192,288,162,324,288,

%T 216,360,528,288,600,432,486,432,840,432,900,768,720,864,864,648,1296,

%U 972,864,1152,1680,648,1764,1440,1296,1584,2208,1152,1764,1800,1728,1728,2808

%N Number of Eisenstein integers in a reduced system modulo n.

%C Equivalent of phi (A000010) in the ring of Eisenstein integers.

%C Number of units in the ring Z[w]/nZ[w], where Z[w] is the ring of Eisenstein integers.

%C a(n) is the number of elements in G(n) = {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.

%C a(n) is the number of ordered pairs (a, b) modulo n such that gcd(a^2 + a*b + b^2, n) = 1.

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

%H Jianing Song, <a href="/A319445/b319445.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>

%F Multiplicative with a(3^e) = 2*3^(2*e-1), a(p^e) = phi(p^e)^2 = (p-1)^2*p^(2*e-2) if p == 1 (mod 3) and J_2(p^e) = A007434(p^e) = (p^2 - 1)*p^(2*e-2) if p == 2 (mod 3).

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (8/27) * Product_{p prime == 1 (mod 3)} (1 - 2/p^2 + 1/p^3) * Product_{p prime == 2 (mod 3)} (1 - 1/p^3) = 0.2410535987... . - _Amiram Eldar_, Feb 13 2024

%e Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.

%e {1, w, w'} is the set of 3 units in the Eisenstein integers modulo 2, so a(2) = 3.

%e {1, w, w^2, -1, w', w'^2} is the set of 6 units in the Eisenstein integers modulo 3, so a(3) = 6.

%e {1, w, w'} is the set of 3 units in the Eisenstein integers modulo 2, so a(2) = 3.

%e {1, w, 1 + w, w', 1 + w', -1 + 2w, -1, -w, -1 - w, -w', -1 - w', -1 + 2w'} is the set of 12 units in the Eisenstein integers modulo 4, so a(4) = 12.

%t f[p_, e_] := If[p == 3 , 2*3^(2*e - 1), Switch[Mod[p, 3], 1, (p - 1)^2*p^(2*e - 2), 2, (p^2 - 1)*p^(2*e - 2)]]; eisPhi[1] = 1; eisPhi[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisPhi, 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*=2*3^(2*e-1));

%o if(p%3==1, r*=(p-1)^2*p^(2*e-2));

%o if(p%3==2, r*=(p^2-1)*p^(2*e-2));

%o );

%o return(r);

%o }

%Y Cf. A007434.

%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), this sequence ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).

%Y Equivalent in the ring of Gaussian integers: A079458.

%K nonn,mult

%O 1,2

%A _Jianing Song_, Sep 19 2018