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%I #50 Sep 24 2022 12:59:36
%S 1,3,7,5,15,31,9,21,63,127,17,51,85,255,73,511,11,33,93,341,1023,23,
%T 89,2047,13,35,39,45,65,91,105,117,195,273,315,455,585,819,1365,4095,
%U 8191,43,129,381,5461,16383,151,217,1057,4681,32767,257,771,1285,3855
%N Triangle T(n,k) of orders of n degree irreducible polynomials over GF(2) listed in ascending order, k=1..A059499(n).
%C 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 Alois P. Heinz, <a href="/A059912/b059912.txt">Rows n = 1..71, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H XIAO Gang, <a href="http://wims.unice.fr/~wims/wims.cgi?session=4GC583E918.2&lang=en&module=tool%2Falgebra%2Fpolyorder.en">Polynomial order: computes the order of an irreducible polynomial over a finite field GF(p)</a>, WIMS.
%F 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 There are 18 (cf. A001037) irreducible polynomials of degree 7 over GF(2) which all have order 127.
%e Triangle T(n,k) begins:
%e 1;
%e 3;
%e 7;
%e 5, 15;
%e 31;
%e 9, 21, 63;
%e 127;
%e 17, 51, 85, 255;
%e 73, 511;
%e 11, 33, 93, 341, 1023;
%e ...
%p with(numtheory):
%p M:= proc(n) option remember;
%p divisors(2^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..20); # _Alois P. Heinz_, May 31 2012
%t 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 (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 Cf. A058943, A059478, A059499, A001037, A059913.
%Y Cf. A212906, A212485, A212486.
%Y Column k=1 of A212737.
%Y Column k=1 gives: A212953.
%Y Last elements of rows give: A000225.
%Y Cf. A108974.
%K easy,nonn,look,tabf
%O 1,2
%A _Vladeta Jovovic_, Feb 09 2001
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