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A235863 Exponent of the multiplicative group G_n:={x+iy: x^2+y^2==1 (mod n); 0 <= x,y < n} where i=sqrt(-1). 7
1, 2, 4, 4, 4, 4, 8, 4, 12, 4, 12, 4, 12, 8, 4, 4, 16, 12, 20, 4, 8, 12, 24, 4, 20, 12, 36, 8, 28, 4, 32, 8, 12, 16, 8, 12, 36, 20, 12, 4, 40, 8, 44, 12, 12, 24, 48, 4, 56, 20, 16, 12, 52, 36, 12, 8, 20, 28, 60, 4, 60, 32, 24, 16, 12, 12, 68, 16, 24, 8, 72 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Jianing Song, Nov 05 2019: (Start)

Exponent of the group G is the least e > 0 such that x^e = 1 for every x in G, where 1 is the identity element.

Also the exponent of O(2,Z_n) or SO(2,Z_n). O(2,Z_n) is the group of 2 X 2 matrices A over Z_n such that A*A^T = E = [1,0;0,1]; SO(2,Z_n) is the group of 2 X 2 matrices A over Z_n such that A*A^T = E = [1,0;0,1] and det(A) = 1. Note that G_n is isomorphic to SO(2,Z_n) by the mapping x+yi <-> [x,y;-y,x]. See A060698 for the group structure of SO(2,Z_n) and A182039 for the group structure of O(2,Z_n). (End)

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..10000

José María Grau, A. M. Oller-Marcén, Manuel Rodriguez and Daniel Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, arXiv:1401.4708 [math.NT], 2014.

José María Grau, A. M. Oller-Marcén, Manuel Rodriguez and Daniel Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, Czechoslovak Mathematical Journal 65(140), (2015) pp. 969-982.

FORMULA

a(@) = 2, a(4) = a(8) = a(16) = 4, a(2^e) = 2^(e-2) for e >= 5; a(p^e) = (p-1)*p^(e-1) if e p == 1 (mod 4) and (p+1)*p^(e-1) if e p == 1 (mod 4). - Jianing Song, Nov 05 2019

If gcd(n,m)=1 then a(nm) = lcm(a(n), a(m)).

MATHEMATICA

fa=FactorInteger; lam[1]=1; lam[p_, s_] := Which[Mod[p, 4] == 3, p ^ (s - 1 ) (p + 1) , Mod[p, 4] == 1, p ^ (s - 1 ) (p - 1)  , s ≥ 5, 2 ^ (s - 2 ), s > 1, 4, s == 1, 2]; lam[n_] := {aux = 1; Do[aux = LCM[aux, lam[fa[n][[i, 1]], fa[n][[i, 2]]]], {i, 1, Length[fa[n]]}]; aux}[[1]] ; Array[lam, 100]

PROG

(PARI) a(n)={my(f=factor(n)); lcm(vector(#f~, i, my([p, e]=f[i, ]); if(p==2, 2^max(e-2, min(e, 2)), p^(e-1)*if(p%4==1, p-1, p+1))))} \\ Andrew Howroyd, Aug 06 2018

CROSSREFS

                                   (Z/nZ)*    ------    G_n

Order:                             A000010    ------  A060968.

Exponent:                          A002322    ------  this sequence.

                                     n-1      ------  A201629.

Carmichael/G-Carmichael numbers:   A002997    ------  A235865.

Lehmer /G-Lehmer numbers:          unknown    ------  A235864.

Cyclic/G-cyclic numbers:           A003277    ------  A235866.

n such that the group is cyclic:   A033948    ------  A235868.

Cf. A060698, A182039.

Sequence in context: A220523 A220527 A183226 * A220495 A194441 A220521

Adjacent sequences:  A235860 A235861 A235862 * A235864 A235865 A235866

KEYWORD

nonn

AUTHOR

José María Grau Ribas, Jan 16 2014

STATUS

approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)