A235866 -id:A235866 - 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)).

A324873 a(n) = gcd(n, A060968(n)).

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, 1, 16, 1, 6, 1, 4, 1, 2, 1, 8, 5, 2, 9, 4, 1, 2, 1, 32, 3, 2, 1, 12, 1, 2, 3, 8, 1, 2, 1, 4, 3, 2, 1, 16, 7, 10, 1, 4, 1, 18, 1, 8, 1, 2, 1, 4, 1, 2, 3, 64, 1, 6, 1, 4, 3, 2, 1, 24, 1, 2, 5, 4, 1, 6, 1, 16, 27, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 5, 32, 1, 14, 9, 20, 1, 2, 1, 8, 1

Offset: 1

Antti Karttunen, May 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A060968, A235866 (positions of ones).

A060968(n) = { my(f=factor(n)[, 1]); n*prod(i=if(n%2, 1, 2), #f, if(f[i]%4==1, 1-1/f[i], 1+1/f[i]))*if(n%4, 1, 2); }; \\ From A060968
A324873(n) = gcd(n,A060968(n));

a(n) = gcd(n, A060968(n)).

A235867 G-cyclic numbers k such that A060968(k)^A060968(k) <> 1 (mod k) and A235863(k)^A235863(k) <> 1 (mod k).

Original entry on oeis.org

77, 119, 133, 187, 217, 253, 287, 301, 319, 323, 341, 391, 399, 403, 407, 413, 437, 469, 517, 551, 553, 559, 583, 589, 623, 651, 667, 707, 713, 731, 737, 749, 779, 781, 803, 817, 851, 869, 871, 889, 893, 899, 903, 913, 917, 935, 943, 959, 969, 1001, 1003

Offset: 1

José María Grau Ribas, Feb 22 2014

nonn

For G-cyclic numbers see A235866.

All terms are composite. - Bill McEachen, Jul 16 2021

Bill McEachen, Table of n, a(n) for n = 1..10000
Jose María Grau, A. M. Oller-Marcen, Manuel Rodriguez and D. Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, arXiv:1401.4708 [math.NT], 2014.

Cf. A235866, A060968, A235863.

genit(maxx)={arr2=List();arr=List();for(ptr=1,maxx,if( gcd(ptr,A060968(ptr))==1,listput(arr,ptr)));for(ptr=2,#arr,n=arr[ptr];a=A060968(n)^A060968(n);b=A235863(n)^A235863(n);if(a%n!=1&&b%n!=1,listput(arr2,n)));}
A060968(n)={my(f=factor(n)[,1]);q=n*prod(i=if(n%2,1,2),#f,if(f[i]%4==1,1-1/f[i],1+1/f[i]))*if(n%4,1,2);return(q);} \\taken from that sequence
A235863(n)={my(f=factor(n));q=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))));return (q);} \\taken from that sequence
\\ Bill McEachen, Jul 16 2021

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A324873 a(n) = gcd(n, A060968(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A235867 G-cyclic numbers k such that A060968(k)^A060968(k) <> 1 (mod k) and A235863(k)^A235863(k) <> 1 (mod k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI