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A059912 Triangle T(n,k) of orders of n degree irreducible polynomials over GF(2) listed in ascending order, k=1..A059499(n).

%I A059912 #51 Feb 16 2025 08:32:44
%S A059912 1,3,7,5,15,31,9,21,63,127,17,51,85,255,73,511,11,33,93,341,1023,23,
%T A059912 89,2047,13,35,39,45,65,91,105,117,195,273,315,455,585,819,1365,4095,
%U A059912 8191,43,129,381,5461,16383,151,217,1057,4681,32767,257,771,1285,3855
%N A059912 Triangle T(n,k) of orders of n degree irreducible polynomials over GF(2) listed in ascending order, k=1..A059499(n).
%C A059912 A permutation of the odd positive numbers; namely, order each odd number d by the multiplicative order of 2 modulo d (in case of a tie, smaller d go first). - _Jeppe Stig Nielsen_, Feb 13 2020
%H A059912 Alois P. Heinz, <a href="/A059912/b059912.txt">Rows n = 1..71, flattened</a>
%H A059912 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H A059912 XIAO Gang, <a href="http://wims.unice.fr/~wims/wims.cgi?session=4GC583E918.2&amp;lang=en&amp;module=tool%2Falgebra%2Fpolyorder.en">Polynomial order: computes the order of an irreducible polynomial over a finite field GF(p)</a>, WIMS.
%F A059912 T(n,k) = k-th smallest element of M(n) = {d : d|(2^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}. - _Alois P. Heinz_, Jun 01 2012
%e A059912 There are 18 (cf. A001037) irreducible polynomials of degree 7 over GF(2) which all have order 127.
%e A059912 Triangle T(n,k) begins:
%e A059912     1;
%e A059912     3;
%e A059912     7;
%e A059912     5,   15;
%e A059912    31;
%e A059912     9,   21,  63;
%e A059912   127;
%e A059912    17,   51,  85, 255;
%e A059912    73,  511;
%e A059912    11,   33,  93, 341, 1023;
%e A059912   ...
%p A059912 with(numtheory):
%p A059912 M:= proc(n) option remember;
%p A059912       divisors(2^n-1) minus U(n-1)
%p A059912     end:
%p A059912 U:= proc(n) option remember;
%p A059912       `if`(n=0, {}, M(n) union U(n-1))
%p A059912     end:
%p A059912 T:= n-> sort([M(n)[]])[]:
%p A059912 seq(T(n), n=1..20);  # _Alois P. Heinz_, May 31 2012
%t A059912 m[n_] := m[n] = Complement[ Divisors[2^n - 1], u[n - 1]]; u[0] = {}; u[n_] := u[n] = Union[ m[n], u[n - 1]]; t[n_, k_] := m[n][[k]]; Flatten[ Table[t[n, k], {n, 1, 16}, {k, 1, Length[ m[n] ]}]] (* _Jean-François Alcover_, Jun 14 2012, after _Alois P. Heinz_ *)
%o A059912 (PARI) maxDegree=26;for(n=1,maxDegree,forstep(d=1,2^n,2,znorder(Mod(2,d))==n&&print1(d,", "))) \\ inefficient, _Jeppe Stig Nielsen_, Feb 13 2020
%Y A059912 Cf. A058943, A059478, A059499, A001037, A059913.
%Y A059912 Cf. A212906, A212485, A212486.
%Y A059912 Column k=1 of A212737.
%Y A059912 Column k=1 gives: A212953.
%Y A059912 Last elements of rows give: A000225.
%Y A059912 Cf. A108974.
%K A059912 easy,nonn,look,tabf
%O A059912 1,2
%A A059912 _Vladeta Jovovic_, Feb 09 2001