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A319445 Number of Eisenstein integers in a reduced system modulo n. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

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

Wikipedia, Eisenstein integer

FORMULA

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).

EXAMPLE

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.

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*=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);

}

CROSSREFS

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.

Sequence in context: A005256 A097979 A215983 * A316318 A173216 A003204

Adjacent sequences:  A319442 A319443 A319444 * A319446 A319447 A319448

KEYWORD

nonn,mult

AUTHOR

Jianing Song, Sep 19 2018

STATUS

approved

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Last modified August 18 20:02 EDT 2019. Contains 326109 sequences. (Running on oeis4.)