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A212486 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order.

%I A212486 #37 Feb 16 2025 08:33:17
%S A212486 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 A212486 32,40,50,60,75,80,96,100,120,150,160,200,240,300,400,480,600,800,
%U A212486 1200,2400,2801,5602,8403,16806,36,43,72,76,86,129,144,152,172,228,258
%N A212486 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order.
%C A212486 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 A212486 R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 560-562.
%D A212486 V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28
%H A212486 Boris Putievskiy, <a href="/A212486/b212486.txt">Table of n, a(n) for n = 1..102</a>
%H A212486 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H A212486 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 A212486 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 A212486 |M(n)| = Sum_{d|n} mu(n/d)*tau(7^d-1) = A059889(n).
%e A212486 Triangle T(n,k) begins:
%e A212486   1,  2,  3,  6;
%e A212486   4,  8, 12, 16, 24,  48;
%e A212486   9, 18, 19, 38, 57, 114, 171, 342;
%e A212486   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 A212486   ...
%p A212486 with(numtheory):
%p A212486 M:= proc(n) option remember;
%p A212486       `if`(n=1, {1, 2, 3, 6}, divisors(7^n-1) minus U(n-1))
%p A212486     end:
%p A212486 U:= proc(n) option remember;
%p A212486       `if`(n=0, {}, M(n) union U(n-1))
%p A212486     end:
%p A212486 T:= n-> sort([M(n)[]])[]:
%p A212486 seq(T(n), n=1..7);
%t A212486 M[n_] := M[n] = If[n == 1, {1, 2, 3, 6}, Divisors[7^n - 1] ~Complement~ U[n - 1]];
%t A212486 U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
%t A212486 T[n_] := Sort[M[n]];
%t A212486 Table[T[n], {n, 1, 7}] // Flatten (* _Jean-François Alcover_, Sep 24 2022, from Maple code *)
%Y A212486 Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.
%Y A212486 Column k=4 of A212737.
%Y A212486 Column k=1 gives: A218358.
%K A212486 easy,nonn,tabf
%O A212486 1,2
%A A212486 _Boris Putievskiy_, Jun 02 2012