A319446 -id:A319446 - cp's OEIS frontend

A319442 Number of divisors of n over the Eisenstein integers.

1, 2, 3, 3, 2, 6, 4, 4, 5, 4, 2, 9, 4, 8, 6, 5, 2, 10, 4, 6, 12, 4, 2, 12, 3, 8, 7, 12, 2, 12, 4, 6, 6, 4, 8, 15, 4, 8, 12, 8, 2, 24, 4, 6, 10, 4, 2, 15, 9, 6, 6, 12, 2, 14, 4, 16, 12, 4, 2, 18, 4, 8, 20, 7, 8, 12, 4, 6, 6, 16, 2, 20, 4, 8, 9, 12, 8, 24, 4, 10

Offset: 1

Jianing Song, Sep 19 2018

Equivalent of d (A000005) in the ring of Eisenstein integers.

Divisors which are associates are identified (two Eisenstein integers z1, z2 are associates if z1 = u * z2 where u is an Eisenstein unit, i.e., one of +-1 or (+-1 +- sqrt(3)*i)/2).

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Divisors of 7 over the Eisenstein integers are 1, 2 + w, 2 + w', 7 and their association, so a(7) = 4.
Divisors of 9 over the Eisenstein integers are 1, 1 + w, 3, 3 + 3w, 9 and their association, so a(9) = 5.

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

Equivalent of arithmetic functions in the ring of Eisenstein integers (the corresponding functions in the ring of integers are in the parentheses): this sequence ("d", A000005), A319449 ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).

Equivalent in the ring of Gaussian integers: A062327.

A319442 := proc(n) local t, f, j, e, m; t := 1: f := ifactors(n)[2];
   for j from 1 to nops(f) do
      e := f[j, 2] + 1; m := f[j, 1] mod 3;
      if   m = 0 then 2*e-1
      elif m = 1 then e^2
      else e fi;
      t := t * % od;
t end: seq(A319442(n), n=1..80); # Peter Luschny, Oct 03 2018

f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisNumDiv, 100] (* Amiram Eldar, Feb 10 2020 *)

A319442(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*=(2*e+1));
        if(p%3==1, r*=(e+1)^2);
        if(p%3==2, r*=(e+1));
    );
    return(r);
}

Multiplicative with a(3^e) = 2*e + 1, a(p^e) = (e + 1)^2 if p == 1 (mod 3) and e + 1 if p == 2 (mod 3).

A319445 Number of Eisenstein integers in a reduced system modulo n.

Original entry on oeis.org

1, 3, 6, 12, 24, 18, 36, 48, 54, 72, 120, 72, 144, 108, 144, 192, 288, 162, 324, 288, 216, 360, 528, 288, 600, 432, 486, 432, 840, 432, 900, 768, 720, 864, 864, 648, 1296, 972, 864, 1152, 1680, 648, 1764, 1440, 1296, 1584, 2208, 1152, 1764, 1800, 1728, 1728, 2808

Offset: 1

Jianing Song, Sep 19 2018

Equivalent of phi (A000010) in the ring of Eisenstein integers.
Number of units in the ring Z[w]/nZ[w], where Z[w] is the ring of Eisenstein integers.
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.
a(n) is the number of ordered pairs (a, b) modulo n such that gcd(a^2 + a*b + b^2, n) = 1.
For n > 2, a(n) is divisible by 6.

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
{1, w, w'} is the set of 3 units in the Eisenstein integers modulo 2, so a(2) = 3.
{1, w, w^2, -1, w', w'^2} is the set of 6 units in the Eisenstein integers modulo 3, so a(3) = 6.
{1, w, w'} is the set of 3 units in the Eisenstein integers modulo 2, so a(2) = 3.
{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.

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

Cf. A007434.
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).
Equivalent in the ring of Gaussian integers: A079458.

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 *)

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

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).

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

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

Original entry on oeis.org

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

José María Grau Ribas, Jul 07 2013

nonn

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)

			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.

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Gaussian integer
Wikipedia, Torsion group

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.

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}]

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

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]

A319443 Number of distinct Eisenstein primes in the factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 3, 2, 1, 1, 2, 2, 2, 3, 2, 1, 2, 1, 3, 1, 3, 1, 3, 2, 1, 2, 2, 3, 2, 2, 3, 3, 2, 1, 4, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 3, 2, 1, 3, 2, 3, 3, 1, 3, 3, 2, 2, 2, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 2, 1, 2, 1, 4, 2, 3, 2

Offset: 1

Jianing Song, Sep 19 2018

nonn

Equivalent of omega (A001221) in the ring of Eisenstein integers.
z is an Eisenstein prime iff z has prime norm or z is the product of a rational prime congruent to 2 modulo 3 and an Eisenstein unit (one of +-1 or (+-1 +- sqrt(3)*i)/2).
Associated Eisenstein prime divisors are counted only once.
Let s(n) be the smallest k with a(k) = n, then we have: s(0) = 1, s(1) = 2, s(2) = 6, s(2n-1) = 2*A121940(n-1), s(2n) = 6*A121940(n-1).

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Over the Gaussian integers, 5187 = 3*7*13*19 is factored as w'*(1 + w)^2*(2 + w)*(2 + w')*(3 + w)*(3 + w')*(3 + 2w)*(3 + 2w'), the distinct Eisenstein prime factors are 1 + w, 2 + w, 2 + w', 3 + w, 3 + w', 3 + 2w and 3 + 2w', so a(5187) = 7.
Over the Gaussian integers, 1006655265000 = 2^3*3^2*5^4*7^5*11^3 is factored as w'^2*(1 + w)^4*2^3*(2 + w)*(2 + w')*5^4*11^3, the distinct Eisenstein prime factors are 1 + w, 2, 2 + w, 2 + w', 5 and 11, so a(1006655265000) = 6.

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

Cf. A121940.
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), A319446 ("psi", A002322), this sequence ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).
Equivalent in the ring of Gaussian integers: A086275.

f[p_, e_] := If[Mod[p, 3] == 1, 2, 1]; eisOmega[1] = 0; eisOmega[n_] := Plus @@ f @@@ FactorInteger[n]; Array[eisOmega, 100] (* Amiram Eldar, Feb 10 2020 *)

a(n)=my(f=factor(n)[, 1]); sum(i=1, #f, if(f[i]%3==1, 2, 1))

Additive with a(p^e) = 2 if p == 1 (mod 3), 1 otherwise.

A319444 Total number of factors in a factorization of n into Eisenstein primes.

Original entry on oeis.org

0, 1, 2, 2, 1, 3, 2, 3, 4, 2, 1, 4, 2, 3, 3, 4, 1, 5, 2, 3, 4, 2, 1, 5, 2, 3, 6, 4, 1, 4, 2, 5, 3, 2, 3, 6, 2, 3, 4, 4, 1, 5, 2, 3, 5, 2, 1, 6, 4, 3, 3, 4, 1, 7, 2, 5, 4, 2, 1, 5, 2, 3, 6, 6, 3, 4, 2, 3, 3, 4, 1, 7, 2, 3, 4, 4, 3, 5, 2, 5, 8, 2, 1, 6, 2, 3, 3

Offset: 1

Jianing Song, Sep 19 2018

nonn

Equivalent of Omega (A001222) in the ring of Eisenstein integers.
z is an Eisenstein prime iff z has prime norm or z is the product of a rational prime congruent to 2 modulo 3 and an Eisenstein unit (one of +-1 or (+-1 +- sqrt(3)*i)/2).
The smallest k with a(k) = n is A038754(n).

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
a(54) = a(2*3^3) = 1*a(2) + 3*a(3) = 1*1 + 3*2 = 7. Over the Gaussian integers, 54 is factored as -2*(1 + w)^6.
a(63) = a(3^2*7) = 2*a(3) + 1*a(7) = 2*2 + 1*2 = 6. Over the Gaussian integers, 63 is factored as w'^2*(1 + w)^4*(2 + w)*(2 + w)'.
a(1006655265000) = a(2^3*3^2*5^4*7^5*11^3) = 3*a(2) + 2*a(3) + 4*a(5) + 5*a(7) + 3*a(11) = 3*1 + 2*2 + 4*1 + 5*2 + 3*1 = 24. Over the Gaussian integers, 1006655265000 is factored as w'^2*(1 + w)^4*2^3*(2 + w)*(2 + w')*5^4*11^3.

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

Cf. A038754.
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), A319446 ("psi", A002322), A319443 ("omega", A001221), this sequence ("Omega", A001222), A319448 ("mu", A008683).
Equivalent in the ring of Gaussian integers: A078458.

f[p_, e_] := e * If[Mod[p, 3] == 2, 1, 2]; eisBigomega[1] = 0; eisBigomega[n_] := Plus @@ f @@@ FactorInteger[n]; Array[eisBigomega, 100] (* Amiram Eldar, Feb 10 2020 *)

a(n)=my(f=factor(n)); sum(i=1, #f~, if(f[i, 1]%3==2, 1, 2)*f[i, 2])

Completely additive with a(p) = 2 if p = 3 or p == 1 (mod 3) and a(p) = 1 if p == 2 (mod 3).

A319448 Moebius function mu(n) defined for the Eisenstein integers.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0

Offset: 1

Jianing Song, Sep 19 2018

Just like the original Moebius function over the integers, a(n) = 0 if n has a squared Eisenstein prime factor, otherwise (-1)^t if n is a product of an Eisenstein unit and t distinct Eisenstein prime factors.

Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2. a(n) = 0 for n divisible by 3 since 3 = w'*(1 + w)^2 contains a squared factor. For rational primes p == 1 (mod 3), p is always factored as (x + y*w)(x + y*w'), x + y*w and x + y*w' are not associated so a(p) = (-1)*(-1) = 1.

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
a(14) = -1 because 14 is factored as 2*(2 + w)*(2 + w') with three distinct Eisenstein prime factors.
a(55) = (-1)*(-1) = 1 because 55 = 5*11 where 5 and 11 are congruent to 2 mod 3 (thus being Eisenstein primes).

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

Cf. A102283.
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), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), this sequence ("mu", A008683).
Equivalent in the ring of Gaussian integers: A318608.

f[p_, e_] := If[p == 3 || e > 1, 0, Switch[Mod[p, 3], 1, 1, 2, -1]]; eisMu[1] = 1; eisMu[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisMu, 100] (* Amiram Eldar, Feb 10 2020 *)

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||e>=2, r=0);
        if(Mod(p, 3)==2&e==1, r*=-1);
    );
    return(r);
}

a(n) = 0 if n is divisible by 3 or has a square prime factor, otherwise Product_{p divides n} (3 - 2*(p mod 3)) where the product is taken over the primes.
Multiplicative with a(p^e) = 0 if p = 3 or e > 1, a(p) = 1 if p == 1 (mod 3) and -1 if p == 2 (mod 3).
For squarefree n, a(n) = Legendre symbol (n, 3) = Kronecker symbol (-3, n) = A102283(n).

A319449 Sum of the norm of divisors of n over Eisenstein integers, with associated divisors counted only once.

Original entry on oeis.org

1, 5, 13, 21, 26, 65, 64, 85, 121, 130, 122, 273, 196, 320, 338, 341, 290, 605, 400, 546, 832, 610, 530, 1105, 651, 980, 1093, 1344, 842, 1690, 1024, 1365, 1586, 1450, 1664, 2541, 1444, 2000, 2548, 2210, 1682, 4160, 1936, 2562, 3146, 2650, 2210, 4433, 3249, 3255

Offset: 1

Jianing Song, Sep 19 2018

Equivalent of sigma (A000203) in the ring of Eisenstein integers. Note that only norms are summed up.

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2, and ||d|| denote the norm of d.
a(3) = ||1|| + ||1 + w|| + ||3|| = 1 + 3 + 9 = 13.
a(7) = ||1|| + ||2 + w|| + ||2 + w'|| + ||7|| = 1 + 7 + 7 + 49 = 64.

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

Cf. A001157.
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), this sequence ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).
Equivalent in the ring of Gaussian integers: A317797.

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

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*=((3^(2*e+1)-1)/2));
        if(Mod(p, 3)==1, r*=((p^(e+1)-1)/(p-1))^2);
        if(Mod(p, 3)==2, r*=(p^(2*e+2)-1)/(p^2-1));
    );
    return(r);
}

Multiplicative with a(3^e) = sigma(3^(2e)) = (3^(2e+1) - 1)/2, a(p^e) = sigma(p^e)^2 = ((p^(e+1) - 1)/(p - 1))^2 if p == 1 (mod 3) and sigma_2(p^e) = A001157(p^e) = (p^(2e+2) - 1)/(p^2 - 1) if p == 2 (mod 3).

A319447 a(n) is the rank of the multiplicative group of Eisenstein integers modulo n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 4, 2, 3, 3, 2, 1, 4, 2, 3, 3, 4, 1, 3, 2, 3, 2, 2, 3, 4, 2, 3, 3, 4, 1, 4, 2, 3, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 5, 3, 2, 1, 4, 2, 3, 5, 3, 3, 3, 2, 3, 2, 4, 1, 4, 2, 3, 2, 4, 3, 4, 2, 4, 3, 2, 1, 5, 2, 3, 2

Offset: 1

Jianing Song, Sep 19 2018

nonn

The rank of a finitely generated group rank(G) is defined to be the size of the minimal generating sets of G.
Let p be an odd prime and (Z[w]/nZ[w])* be the multiplicative group of Gaussian integers modulo n, then: (Z[w]/p^e*Z[w])* = (C_((p-1)*p^(e-1)))^2 if p == 1 (mod 6); C_(p^(e-1)) X C_(p^(e-1)*(p^2-1)) if p == 5 (mod 6); (Z[w]/3^e*Z[w])* = C_3 X C_(3^(e-1)) X C_(2*3^(e-1)); (Z[w]/2Z[w])* = C_3, (Z[w]/2^e*Z[w])* = C_2 X C_(2^(e-2)) X C_(3*2^(e-1)) for e >= 2. If n = Product_{i=1..k} (p_i)^(e_i), then (Z[w]/nZ[w])* = (Z[w]/(p_1)^(e_1)*Z[w])* X (Z[w]/(p_2)^(e_2)*Z[w])* X ... X (Z[w]/(p_k)^(e_k)*Z[w])*.
The order of (Z[w]/nZ[w])* is A319445(n) and the exponent of it is A319446(n).
{a(n)} is not additive: (Z[w]/2Z[w])* = C_3, (Z[w]/25Z[w])* = C_5 X C_120, so (Z[w]/50Z[w])* = C_15 X C_120, a(50) < a(2) + a(25).
A319445(n)/A319446(n) is always an integer, and is 1 if and only if (Z[w]/nZ[w])* is cyclic, that is, rank((Z[w]/nZ[w])*) = a(n) = 0 or 1, and n has a primitive root in (Z[w]/nZ[w])*. a(n) = 1 if and only if n = 3 or a prime congruent to 2 mod 3. - Jianing Song, Jan 08 2019
From Jianing Song, Oct 03 2022: (Start)
More generally, let pi be a prime element of Z[w] of norm p or p^2 for prime p, then:
- for p == 1 (mod 6), (Z[w]/(pi^e)Z[w])* = C_((p-1)*p^(e-1));
- for p == 5 (mod 6), (Z[w]/(pi^e)Z[w])* = C_(p^(e-1)) X C_(p^(e-1)*(p^2-1));
- for p = 3, (Z[w]/(pi^e)Z[w])* = C_2 for e = 1, C_3 X C_(3^floor((e-2)/2)) X C_(2*3^ceiling((e-2)/2)) for e >= 2;
- for p = 2, (Z[w]/(pi^e)Z[w])* = C_3 for e = 1, C_2 X C_(2^(e-2)) X C_(3*2^(e-1)) for e >= 2.
For a more general result see my link below. (End)

			(Z[w]/1Z[w])* = C_1 (has rank 0);
(Z[w]/2Z[w])* = C_3 (has rank 1);
(Z[w]/3Z[w])* = C_6 (has rank 1);
(Z[w]/4Z[w])* = C_2 X C_6 (has rank 2);
(Z[w]/5Z[w])* = C_24 (has rank 1);
(Z[w]/6Z[w])* = C_3 X C_6 (has rank 2);
(Z[w]/7Z[w])* = C_6 X C_6 (has rank 2);
(Z[w]/8Z[w])* = C_2 X C_2 X C_12 (has rank 3);
(Z[w]/9Z[w])* = C_3 X C_3 X C_6 (has rank 3);
(Z[w]/10Z[w])* = C_3 X C_24 (has rank 2).

Jianing Song, Table of n, a(n) for n = 1..10000
Jianing Song, Structure of (R/P^e)* for R the ring of integers of a quadratic field and P a prime ideal of R
Wikipedia, Eisenstein integer

Cf. A046072, A319445, A319446.

Equivalent in the ring of Gaussian integers: A316506.

rad(n) = factorback(factorint(n)[, 1]);
grad(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==1, r*=3);
        if(p==2&e==2, r*=12);
        if(p==2&e>=3, r*=24);
        if(p==3&e==1, r*=6);
        if(p==3&e>=2, r*=54);
        if(p%6==1, r*=(rad(p-1))^2);
        if(p%6==5&e==1, r*=rad(p^2-1));
        if(p%6==5&e>=2, r*=p^2*rad(p^2-1));
    );
    return(r);
}
a(n)=if(n>1, vecmax(factor(grad(n))[, 2]), 0); \\ The program is based on the facts that although rank((Z[w]/nZ[w])*) is not additive, the p-rank of (Z[w]/nZ[w])* is additive for any prime p, and that rank((Z[w]/nZ[w])*) is the maximum of the p-rank of (Z[w]/nZ[w])* where p runs through all primes. - Jianing Song, Aug 05 2019

Let p be an odd prime, then: a(p^e) = 2 if p == 1 (mod 6) or p == 5 (mod 6), e >= 2; a(p) = 1 if p == 5 (mod 6). a(3) = 1, a(3^e) = 3 for e >= 2. a(2) = 1, a(4) = 2, a(2^e) = 3 for e >= 3. [Corrected by Jianing Song, Aug 05 2019]

Corrected by Jianing Song, Jan 12 2019

cp's OEIS Frontend

A319442 Number of divisors of n over the Eisenstein integers.

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A319445 Number of Eisenstein integers in a reduced system modulo n.

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

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A319443 Number of distinct Eisenstein primes in the factorization of n.

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A319444 Total number of factors in a factorization of n into Eisenstein primes.

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A319448 Moebius function mu(n) defined for the Eisenstein integers.

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A319449 Sum of the norm of divisors of n over Eisenstein integers, with associated divisors counted only once.

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