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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 January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)