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

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A212486 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order. 6

%I #36 Sep 24 2022 13:14:35

%S 1,2,3,6,4,8,12,16,24,48,9,18,19,38,57,114,171,342,5,10,15,20,25,30,

%T 32,40,50,60,75,80,96,100,120,150,160,200,240,300,400,480,600,800,

%U 1200,2400,2801,5602,8403,16806,36,43,72,76,86,129,144,152,172,228,258

%N Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order.

%C The elements m of row n, are also solutions to the equation: multiplicative order of 7 mod m = n, with gcd(m,7) = 1, cf. A053450.

%D R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 560-562.

%D V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

%H Boris Putievskiy, <a href="/A212486/b212486.txt">Table of n, a(n) for n = 1..102</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>

%H Gang Xiao, <a href="http://wims.unice.fr/wims/wims.cgi?session=4GC583E918.2&amp;lang=en&amp;module=tool%2Falgebra%2Fpolyorder.en">(computes the order of an irreducible polynomial over a finite field GF(p))</a>

%F T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (7^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,3,6}.

%F |M(n)| = Sum_{d|n} mu(n/d)*tau(7^d-1) = A059889(n).

%e Triangle T(n,k) begins:

%e 1, 2, 3, 6;

%e 4, 8, 12, 16, 24, 48;

%e 9, 18, 19, 38, 57, 114, 171, 342;

%e 5, 10, 15, 20, 25, 30, 32, 40, 50, 60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800, 1200, 2400;

%e ...

%p with(numtheory):

%p M:= proc(n) option remember;

%p `if`(n=1, {1, 2, 3, 6}, divisors(7^n-1) minus U(n-1))

%p end:

%p U:= proc(n) option remember;

%p `if`(n=0, {}, M(n) union U(n-1))

%p end:

%p T:= n-> sort([M(n)[]])[]:

%p seq(T(n), n=1..7);

%t M[n_] := M[n] = If[n == 1, {1, 2, 3, 6}, Divisors[7^n - 1] ~Complement~ U[n - 1]];

%t U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];

%t T[n_] := Sort[M[n]];

%t Table[T[n], {n, 1, 7}] // Flatten (* _Jean-François Alcover_, Sep 24 2022, from Maple code *)

%Y Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.

%Y Column k=4 of A212737.

%Y Column k=1 gives: A218358.

%K easy,nonn,tabf

%O 1,2

%A _Boris Putievskiy_, Jun 02 2012

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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 06:56 EDT 2024. Contains 370953 sequences. (Running on oeis4.)