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 A212906 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(3) listed in ascending order. 7
 1, 2, 4, 8, 13, 26, 5, 10, 16, 20, 40, 80, 11, 22, 121, 242, 7, 14, 28, 52, 56, 91, 104, 182, 364, 728, 1093, 2186, 32, 41, 82, 160, 164, 205, 328, 410, 656, 820, 1312, 1640, 3280, 6560, 757, 1514, 9841, 19682, 44, 61, 88, 122, 244, 484, 488, 671, 968, 1342 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The elements m of row n, are also solutions to the equation: multiplicative order of 3 mod m = n, with gcd(m,3) = 1, cf. A053446. REFERENCES R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 555-557. V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28 LINKS Boris Putievskiy and Alois P. Heinz, Rows n = 1..47, flattened (first 13 rows from Boris Putievskiy) Eric Weisstein's World of Mathematics, Irreducible Polynomial XIAO, Polynomial order (computes the order of an irreducible polynomial over a finite field GF(p)) FORMULA T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (3^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2}. |M(n)| = Sum_{d|n} mu(n/d)*tau(3^d-1) = A059885(n). EXAMPLE Triangle T(n,k) begins: 1,   2; 4,   8; 13, 26; 5,  10,  16,  20, 40, 80; 11, 22, 121, 242; 7,  14,  28,  52, 56, 91, 104, 182, 364, 728; MAPLE with(numtheory): M:= proc(n) option remember;       divisors(3^n-1) minus U(n-1)     end: U:= proc(n) option remember;       `if`(n=0, {}, M(n) union U(n-1))     end: T:= n-> sort([M(n)[]])[]: seq(T(n), n=1..15);  # Alois P. Heinz, Jun 02 2012 MATHEMATICA M[n_] := M[n] = Divisors[3^n - 1] ~Complement~ U[n - 1]; U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]]; T[n_] := Sort[M[n]]; Array[T, 15] // Flatten (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *) CROSSREFS Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892. Column k=2 of A212737. Column k=1 gives: A218356. Sequence in context: A248876 A102704 A196720 * A043774 A043777 A043781 Adjacent sequences:  A212903 A212904 A212905 * A212907 A212908 A212909 KEYWORD easy,nonn,look,tabf AUTHOR Boris Putievskiy, May 29 2012 STATUS approved

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Last modified December 14 23:10 EST 2018. Contains 318141 sequences. (Running on oeis4.)