The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A328643 Positive integers m such that the matrix E_m has order 3^m-1 in GL_m(3) where E_m is the m X m invertible tridiagonal matrix with all nonzero entries equal to 1 except for the (m,m) entry that is equal to 2. 1
 1, 3, 5, 9, 11, 23, 29, 35, 39, 41, 53, 65, 69, 81, 83, 89, 95, 99, 105, 113, 119, 131, 155, 173, 179, 189, 191, 209, 221, 231, 233, 239, 243, 251, 281, 293, 299, 303, 323, 329, 359, 371, 375, 411, 413, 419, 429, 431, 443, 453, 491 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The cyclic subgroups of GL_m(q) of order q^m-1 are called Singer cycles. LINKS M. Farrokhi D. G., Lattice paths inside a table: Rows and columns linear combinations, arXiv:1910.09844 [math.CO], 2019. W. M. Kantor, Linear groups containing a Singer cycle, J. Algebra 62(1) (1980), 232-234. EXAMPLE For n = 3 the a(3) = 5 solution is the matrix E_5 = [ [ 1 1 0 0 0 ], [ 1 1 1 0 0 ], [ 0 1 1 1 0 ], [ 0 0 1 1 1 ], [ 0 0 0 1 2 ] ] since the matrix E_5 has order 3^5 - 1 = 242 in GL_5(3). PROG (GAP) EMatrix := function(n, q) local M, i; M := NullMat(n, n, GF(q)); for i in [2..n] do M[i - 1][i - 1] := Z(q) ^ 0; M[i - 1][i] := Z(q) ^ 0; M[i][i - 1] := Z(q) ^ 0; od; M[n][n] := 2 * Z(q) ^ 0; return M; end; for n in [1..100] do M := EMatrix(n, 3); if Determinant(M) <> 0 * Z(3) and Order(M) = 3 ^ n - 1 then Print(n, "\n"); fi; od; (PARI) E(m)={matrix(m, m, i, j, (i==m&&j==m) + (abs(i-j)<=1))} is(m, b)={my(ID=matid(m), M=Mod(E(m), b), e=b^m-1); if(M^e==ID, fordiv(e, d, if(d2&&is(m, 3), print1(m, ", "))) \\ Andrew Howroyd, Dec 21 2019 CROSSREFS Cf. A328642. Sequence in context: A092917 A256220 A163778 * A160358 A319084 A120806 Adjacent sequences: A328640 A328641 A328642 * A328644 A328645 A328646 KEYWORD nonn AUTHOR M. Farrokhi D. G., Oct 23 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 28 04:25 EST 2023. Contains 359850 sequences. (Running on oeis4.)