A316506 -id:A316506 - cp's OEIS frontend

A323019 a(n) is the smallest k such that A316506(k) = n.

1, 2, 4, 8, 20, 40, 120, 520, 1560, 8840, 26520, 185640, 769080, 5383560, 28455960, 199191720, 1166694360, 8166860520, 61834801080, 432843607560, 3771922865880, 26403460061160, 275350369209240, 1927452584464680, 21201978429111480, 171543280017356520

Offset: 0

Jianing Song, Jan 10 2019

nonn

a(n) is the smallest k such that the rank of the multiplicative group of Gaussian integers modulo k is n.

			a(2) = 4, i(2) = 0, j(2) = 0;
a(3) = 8, i(3) = 0, j(3) = 0;
For n = 4, a(n-2)*p(i(n-2)+1) = a(2)*p(1) = 4*5 = 20, a(n-1)*q(j(n-1)+1) = a(3)*q(1) = 8*3 = 24. So a(4) = 20, i(4) = i(2) + 1 = 1, j(4) = j(2) = 0.
For n = 5, a(n-2)*p(i(n-2)+1) = a(3)*p(1) = 8*5 = 40, a(n-1)*q(j(n-1)+1) = a(4)*q(1) = 20*3 = 60. So a(5) = 40, i(5) = i(3) + 1 = 1, j(5) = j(3) = 0.
For n = 6, a(n-2)*p(i(n-2)+1) = a(4)*p(2) = 20*13 = 260, a(n-1)*q(j(n-1)+1) = a(5)*q(1) = 40*3 = 120. So a(6) = 120, i(6) = i(5) = 1, j(6) = j(5) + 1 = 1.
...
List of the multiplicative groups of Gaussian integers modulo members of this sequence:
a(0) = 1: the trivial group;
a(1) = 2: C_2;
a(2) = 4: C_2 X C_4;
a(3) = 8: C_2 X C_4 X C_4;
a(4) = 20: C_2 X C_4 X C_4 X C_4;
a(5) = 40: C_2 X C_4 X C_4 X C_4 X C_4;
a(6) = 120: C_2 X C_4 X C_4 X C_4 X C_4 X C_8;
a(7) = 520: C_2 X C_4 X C_4 X C_4 X C_4 X C_12 X C_12;
a(8) = 1560: C_2 X C_4 X C_4 X C_4 X C_4 X C_4 X C_12 X C_24;
a(9) = 8840: C_2 X C_4 X C_4 X C_4 X C_4 X C_4 X C_4 X C_48 X C_48;
a(10) = 26520: C_2 X C_4 X C_4 X C_4 X C_4 X C_4 X C_4 X C_8 X C_48 X C_48;
...

Cf. A316506.

PARI
```
p(n) = my(i=0, k=0); while(i
				
```

a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8. Let p(n) be the n-th prime congruent to 1 modulo 4, q(n) be the n-th prime congruent to 3 modulo 4. Then there exists {i(n)} and {j(n)} such that i(2) = j(2) = i(3) = j(3) = 0; for n >= 4, if a(n-2)*p(i(n-2)+1) < a(n-1)*q(j(n-1)+1), then a(n) = a(n-2)*p(i(n-2)+1), i(n) = i(n-2) + 1, j(n) = j(n-2), or a(n) = a(n-1)*q(j(n-1)+1), i(n) = i(n-1), j(n) = j(n-1) + 1.

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]

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

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A323019 a(n) is the smallest k such that A316506(k) = n.

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

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

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