cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A323020 a(n) is the smallest k such that A319447(k) = n.

Original entry on oeis.org

1, 2, 4, 8, 18, 56, 126, 630, 1638, 8190, 31122, 155610, 964782, 4823910, 35696934, 178484670, 1534968162, 7674840810, 84423248910, 468165289410, 5149818183510, 31367074390470, 345037818295170, 2289796430504310, 25187760735547410, 180893918009840490
Offset: 0

Views

Author

Jianing Song, Jan 10 2019

Keywords

Comments

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

Examples

			a(3) = 8, i(3) = 0, j(3) = 0;
a(4) = 18, i(4) = 0, j(4) = 0;
For n = 5, a(n-2)*p(i(n-2)+1) = a(3)*p(1) = 8*7 = 56, a(n-1)*q(j(n-1)+1) = a(4)*q(1) = 18*5 = 90. So a(5) = 56, 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(1) = 18*7 = 126, a(n-1)*q(j(n-1)+1) = a(5)*q(1) = 56*5 = 280. So a(6) = 126, i(6) = i(4) + 1 = 2, j(6) = j(4) = 0.
For n = 7, a(n-2)*p(i(n-2)+1) = a(5)*p(2) = 56*13 = 728, a(n-1)*q(j(n-1)+1) = a(6)*q(1) = 126*5 = 630. So a(7) = 630, i(7) = i(6) = 1, j(7) = j(6) + 1 = 1.
...
List of the multiplicative groups of Eisenstein integers modulo members of this sequence:
a(0) = 1: the trivial group;
a(1) = 2: C_3;
a(2) = 4: C_2 X C_6;
a(3) = 8: C_2 X C_2 X C_12;
a(4) = 18: C_3 X C_3 X C_3 X C_6;
a(5) = 56: C_2 X C_2 X C_6 X C_6 X C_12;
a(6) = 126: C_3 X C_3 X C_3 X C_6 X C_6 X C_6;
a(7) = 630: C_3 X C_3 X C_3 X C_6 X C_6 X C_6 X C_24;
a(8) = 1638: C_3 X C_3 X C_3 X C_6 X C_6 X C_6 X C_12 X C_12;
a(9) = 8190: C_3 X C_3 X C_3 X C_6 X C_6 X C_6 X C_6 X C_6 X C_24;
a(10) = 31122: C_3 X C_3 X C_3 X C_6 X C_6 X C_6 X C_6 X C_6 X C_36 X C_36;
...

Crossrefs

Cf. A319447.

Programs

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

Formula

a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, a(4) = 18. Let p(n) be the n-th prime congruent to 1 modulo 6, q(n) be the n-th prime congruent to 5 modulo 6. Then there exists {i(n)} and {j(n)} such that i(3) = j(3) = i(4) = j(4) = 0; for n >= 5, 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.

A319446 Exponent of the group of the Eisenstein integers in a reduced system modulo n.

Original entry on oeis.org

1, 3, 6, 6, 24, 6, 6, 12, 6, 24, 120, 6, 12, 6, 24, 24, 288, 6, 18, 24, 6, 120, 528, 12, 120, 12, 18, 6, 840, 24, 30, 48, 120, 288, 24, 6, 36, 18, 12, 24, 1680, 6, 42, 120, 24, 528, 2208, 24, 42, 120, 288, 12, 2808, 18, 120, 12, 18, 840, 3480, 24, 60
Offset: 1

Views

Author

Jianing Song, Sep 19 2018

Keywords

Comments

Equivalent of psi (A002322) in the ring of Eisenstein integers.
a(n) is the exponent of the multiplicative group of Eisenstein integers modulo n, i.e., (Z[w]/nZ[w])* = {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. The number of elements in (Z[w]/nZ[w])* is A319445(n).
a(n) is the smallest e such that for any Eisenstein integer z coprime to n we have z^e == 1 (mod n).
By definition, A319445(n)/a(n) is always an integer, and is 1 iff (Z[w]/nZ[w])* is cyclic, that is, rank((Z[w]/nZ[w])*) = A319447(n) = 0 or 1, and n has a primitive root in (Z[w]/nZ[w])*. A319445(n)/a(n) = 1 iff n = 1, 3 or a prime congruent to 2 mod 3.
For n > 2, a(n) is divisible by 6.

Examples

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Let G = (Z[w]/4Z[w])* = {1, w, 1 + w, w', 1 + w', -1 + 2w, -1, -w, -1 - w, -w', -1 - w', -1 + 2w'}. The possibilities for the exponent of G are 12, 6, 4, 3, 2 and 1. G^6 = {x^6 mod 4 : x belongs to G} = {1} and w^3 !== 1 (mod 4), w^4 !== 1 (mod 4). Therefore, the exponent of G is greater than 4, accordingly the exponent of G is 6 and a(4) = 6.

Links

Crossrefs

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), this sequence ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).
Equivalent in the ring of Gaussian integers: A227334.

Programs

  • Mathematica
    f[p_, e_] := If[p == 3 , If[e == 1, 6, 2*3^(e - 1)], Switch[Mod[p, 3], 1, (p - 1)*p^(e - 1), 2, (p^2 - 1)*p^(e - 1)]]; eisPsi[1] = 1; eisPsi[n_] := LCM @@ f @@@ FactorInteger[n]; Array[eisPsi, 100]  (* Amiram Eldar, Feb 10 2020 *)
  • 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=lcm(r,2*3^max(e-1,1)));
        if(p%3==1, r=lcm(r,(p-1)*p^(e-1)));
        if(p%3==2, r=lcm(r,(p^2-1)*p^(e-1)));
    );
    return(r);
}

Formula

a(3) = 6, a(3^e) = 2*3^(e-1) for e >= 2; a(p^e) = (p - 1)*p^(e-1) if p == 1 (mod 3) and (p^2 - 1)*p^(e-1) if p == 2 (mod 3). If gcd(m, n) = 1 then a(mn) = lcm(a(m), a(n)). [See the group structure of (Z[w]/(pi^e)Z[w])* in A319447, where pi is a prime element in Z[w]. - Jianing Song, Oct 03 2022]

Extensions

Corrected by Jianing Song, Jan 12 2019

A316506 a(n) is the rank of the multiplicative group of Gaussian integers modulo n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 3, 2, 3, 1, 3, 2, 2, 3, 3, 2, 2, 1, 4, 2, 2, 1, 4, 2, 3, 2, 3, 2, 4, 1, 3, 2, 3, 3, 3, 2, 2, 3, 5, 2, 3, 1, 3, 3, 2, 1, 4, 2, 3, 3, 4, 2, 2, 3, 4, 2, 3, 1, 5, 2, 2, 3, 3, 4, 3, 1, 4, 2, 4, 1, 4, 2, 3, 3, 3, 2, 4, 1, 5, 2, 3, 1, 4, 4, 2, 3
Offset: 1

Views

Author

Jianing Song, Jul 05 2018

Keywords

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[i]/nZ[i])* be the multiplicative group of Gaussian integers modulo n, then: (Z[i]/p^e*Z[i])* = (C_((p-1)*p^(e-1)))^2 if p == 1 (mod 4); C_(p^(e-1)) X C_(p^(e-1)*(p^2-1)) if p == 3 (mod 4). (Z[i]/2Z[i])* = C_2, (Z[i]/2^e*Z[i])* = C_4 X C_(2^(e-2)) X C_(2^(e-1)) for e >= 2. If n = Product_{i=1..k} (p_i)^(e_i), then (Z[i]/nZ[i])* = (Z[i]/(p_1)^(e_1)*Z[i])* X (Z[i]/(p_2)^(e_2)*Z[i])* X ... X (Z[i]/(p_k)^(e_k)*Z[i])*.
The order of (Z[i]/nZ[i])* is A079458(n) and the exponent of it is A227334(n).
{a(n)} is not additive: (Z[i]/2Z[i])* = C_2, (Z[i]/9Z[i])* = C_3 X C_24, so (Z[i]/18Z[i])* = C_6 X C_24, a(18) < a(2) + a(9). The same problem occurs for a(36), a(54) and a(72) and so on. But note that (Z[i]/63Z[i])* = C_3 X C_24 X C_48 and a(63) = a(7) + a(9).
A079458(n)/A227334(n) is always an integer, and is 1 if and only if (Z[i]/nZ[i])* is cyclic, that is, rank((Z[i]/nZ[i])*) = a(n) = 0 or 1, and n has a primitive root in (Z[i]/nZ[i])*. a(n) = 1 if and only if n = 2 or a prime congruent to 3 mod 4. - Jianing Song, Jan 08 2019
From Jianing Song, Oct 03 2022: (Start)
More generally, let pi be a prime element of Z[i] of norm p or p^2 for prime p, then:
- for p == 1 (mod 4), (Z[i]/(pi^e)Z[i])* = C_((p-1)*p^(e-1));
- for p == 3 (mod 4), (Z[i]/(pi^e)Z[i])* = C_(p^(e-1)) X C_(p^(e-1)*(p^2-1));
- for p = 2, (Z[i]/(pi^e)Z[i])* = C_1 for e = 1, C_2 for e = 2, C_4 X C_(2^floor((e-3)/2)) X C_(2^ceiling((e-3)/2)) for e >= 3.
For a more general result see my link below. (End)

Examples

			(Z[i]/1Z[i])* = C_1 (has rank 0);
(Z[i]/2Z[i])* = C_2 (has rank 1);
(Z[i]/3Z[i])* = C_8 (has rank 1);
(Z[i]/4Z[i])* = C_2 X C_4 (has rank 2);
(Z[i]/5Z[i])* = C_4 X C_4 (has rank 2);
(Z[i]/6Z[i])* = C_2 X C_8 (has rank 2);
(Z[i]/7Z[i])* = C_48 (has rank 1);
(Z[i]/8Z[i])* = C_2 X C_4 X C_4 (has rank 3);
(Z[i]/9Z[i])* = C_3 X C_24 (has rank 2);
(Z[i]/10Z[i])* = C_2 X C_4 X C_4 (has rank 3).

Links

Crossrefs

Cf. A046072, A079458, A227334, A302254.
Equivalent in the ring of Eisenstein integers: A319447.

Programs

  • 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*=2);
        if(p==2&e==2, r*=4);
        if(p==2&e>=3, r*=8);
        if(p%4==1, r*=(rad(p-1))^2);
        if(p%4==3&e==1, r*=rad(p^2-1));
        if(p%4==3&e>=2, r*=p^2*rad(p^2-1));
    );
    return(r);
}
a(n)=if(n>1, vecmax(factor(grad(n))[, 2]), 0);

Formula

Let p be an odd prime, then: a(p^e) = 2 if p == 1 (mod 4) or p == 3 (mod 4), e >= 2; a(p) = 1 if p == 3 (mod 4). a(2) = 1, a(4) = 2, a(2^e) = 3 for e >= 3.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.