login
Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(23) listed in ascending order.
4

%I #24 Feb 12 2023 06:14:17

%S 1,2,11,22,3,4,6,8,12,16,24,33,44,48,66,88,132,176,264,528,7,14,77,79,

%T 154,158,553,869,1106,1738,6083,12166,5,10,15,20,30,32,40,53,55,60,80,

%U 96,106,110,120,159,160,165,212,220,240,265,318,330,352,424,440

%N Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(23) listed in ascending order.

%H Alois P. Heinz, <a href="/A218340/b218340.txt">Rows n = 1..17, flattened</a>

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

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

%F T(n,k) = k-th smallest element of M(n) = {d : d|(23^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

%e Triangle begins:

%e 1, 2, 11, 22;

%e 3, 4, 6, 8, 12, 16, 24, 33, 44, ...

%e 7, 14, 77, 79, 154, 158, 553, 869, 1106, ...

%e 5, 10, 15, 20, 30, 32, 40, 53, 55, ...

%e 292561, 585122, 3218171, 6436342;

%e ...

%p with(numtheory):

%p M:= proc(n) M(n):= divisors(23^n-1) minus U(n-1) end:

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

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

%p seq(T(n), n=1..5);

%t M[n_] := M[n] = Divisors[23^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, 5}] // Flatten (* _Jean-François Alcover_, Feb 12 2023, after _Alois P. Heinz_ *)

%Y Column k=9 of A212737.

%Y Column k=1 gives: A218363.

%Y Row lengths are A212957(n,23).

%K nonn,tabf,look

%O 1,2

%A _Alois P. Heinz_, Oct 26 2012