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) = 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 p == 1 (mod 4) and (p+1)*p^(e-1) if 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
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Jan 16 2014
STATUS
approved