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A319447 a(n) is the rank of the multiplicative group of Eisenstein integers modulo n. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

LINKS

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

Wikipedia, Eisenstein integer

FORMULA

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]

EXAMPLE

(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).

PROG

(PARI)

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

CROSSREFS

Cf. A046072, A319445, A319446.

Equivalent in the ring of Gaussian integers: A316506.

Sequence in context: A129193 A262446 A205147 * A224961 A056812 A210721

Adjacent sequences:  A319444 A319445 A319446 * A319448 A319449 A319450

KEYWORD

nonn

AUTHOR

Jianing Song, Sep 19 2018

EXTENSIONS

Corrected by Jianing Song, Jan 12 2019

STATUS

approved

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Last modified January 22 13:41 EST 2020. Contains 331149 sequences. (Running on oeis4.)