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Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(29) listed in ascending order.
4

%I #20 Feb 12 2023 06:14:13

%S 1,2,4,7,14,28,3,5,6,8,10,12,15,20,21,24,30,35,40,42,56,60,70,84,105,

%T 120,140,168,210,280,420,840,13,26,52,67,91,134,182,268,364,469,871,

%U 938,1742,1876,3484,6097,12194,24388,16,48,80,112,240,336,421,560

%N Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(29) listed in ascending order.

%H Alois P. Heinz, <a href="/A218341/b218341.txt">Rows n = 1..16, 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|(29^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, 4, 7, 14, 28;

%e : 3, 5, 6, 8, 10, 12, 15, ...

%e : 13, 26, 52, 67, 91, 134, 182, ...

%e : 16, 48, 80, 112, 240, 336, 421, ...

%e : 732541, 1465082, 2930164, 5127787, 10255574, 20511148;

%p with(numtheory):

%p M:= proc(n) M(n):= divisors(29^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[29^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=10 of A212737.

%Y Column k=1 gives: A218364.

%Y Row lengths are A212957(n,29).

%K nonn,tabf,look

%O 1,2

%A _Alois P. Heinz_, Oct 26 2012