A235868 -id:A235868 - cp's OEIS frontend

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).

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

Offset: 1

José María Grau Ribas, Jan 16 2014

nonn

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)

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.

                                   (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.

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]

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

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)).

A100836 a(n) is the smallest value k > 1 such that k^2 - 1 is divisible by n^2.

Original entry on oeis.org

2, 3, 8, 7, 24, 17, 48, 31, 80, 49, 120, 17, 168, 97, 26, 127, 288, 161, 360, 49, 197, 241, 528, 127, 624, 337, 728, 97, 840, 199, 960, 511, 485, 577, 99, 161, 1368, 721, 170, 351, 1680, 197, 1848, 241, 649, 1057, 2208, 127, 2400, 1249, 577, 337, 2808, 1457, 1451

Offset: 1

Thomas Kerscher (Thomas.Kerscher(AT)web.de), Jan 19 2005

a(n) = n^2 - 1 if n > 1 is in A235868. - Robert Israel, Jan 17 2019

			a(4)=7 because 7^2 - 1 is divisible by 4^2 (and 7 is the smallest integer > 1 that satisfies this criterion).

Robert Israel, Table of n, a(n) for n = 1..10000 (first 500 terms from Harvey P. Dale)

Cf. A235868.

f:= n -> min(map(t -> rhs(op(t)),{msolve(k^2-1,n^2)}) minus {1}):
f(1):= 2:
map(f, [$1..100]); # Robert Israel, Jan 17 2019

With[{c=Range[2,10000]},Flatten[Table[Select[c,Divisible[#^2-1, n^2]&, 1],{n,60}]]] (* Harvey P. Dale, Oct 23 2011 *)

{ A100836(n)=local(f,b,t,m); if(n==1,return(1)); if(n==2,return(3));t=valuation(n,2); if(n==2^t, return(2^(2*t-1)-1)); f=factorint(n/2^t);f=vector(matsize(f)[1],j,f[j,1]^(2*f[j,2])); if(t>0, f=concat(f,[2^(2*t-1)])); b=n^2+1; forvec(v=vector(#f,i,[0,1]), m=lift(chinese(vector(#f,j,Mod((-1)^v[j],f[j])))); if(m>1, b=min(b,m)); ); b } /* Max Alekseyev, Nov 21 2008 */

Entries confirmed by Ray Chandler, R. J. Mathar, and Max Alekseyev, Nov 21 2008

cp's OEIS Frontend

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).

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