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A218339 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(19) listed in ascending order.

%I A218339 #26 Feb 18 2025 08:12:59
%S A218339 1,2,3,6,9,18,4,5,8,10,12,15,20,24,30,36,40,45,60,72,90,120,180,360,
%T A218339 27,54,127,254,381,762,1143,2286,3429,6858,16,48,80,144,181,240,362,
%U A218339 543,720,724,905,1086,1448,1629,1810,2172,2715,2896,3258,3620,4344,5430
%N A218339 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(19) listed in ascending order.
%H A218339 Alois P. Heinz, <a href="/A218339/b218339.txt">Rows n = 1..18, flattened</a>
%H A218339 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H A218339 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolynomialOrder.html">Polynomial Order</a>
%F A218339 T(n,k) = k-th smallest element of M(n) = {d : d|(19^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.
%e A218339 Triangle begins:
%e A218339     1,   2,   3,   6,   9,   18;
%e A218339     4,   5,   8,  10,  12,   15,   20,   24,   30,   36,   40, ...
%e A218339    27,  54, 127, 254, 381,  762, 1143, 2286, 3429, 6858;
%e A218339    16,  48,  80, 144, 181,  240,  362,  543,  720,  724,  905, ...
%e A218339   151, 302, 453, 906, 911, 1359, 1822, 2718, 2733, 5466, 8199, ...
%e A218339   ...
%p A218339 with(numtheory):
%p A218339 M:= proc(n) M(n):= divisors(19^n-1) minus U(n-1) end:
%p A218339 U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
%p A218339 T:= n-> sort([M(n)[]])[]:
%p A218339 seq(T(n), n=1..5);
%t A218339 M[n_] := M[n] = Divisors[19^n-1] ~Complement~ U[n-1];
%t A218339 U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]];
%t A218339 T[n_] := Sort[M[n]];
%t A218339 Table[T[n], {n, 1, 5}] // Flatten (* _Jean-François Alcover_, Feb 12 2023, after _Alois P. Heinz_ *)
%Y A218339 Column k=8 of A212737.
%Y A218339 Column k=1 gives: A218362.
%Y A218339 Row lengths are A212957(n,19).
%K A218339 nonn,tabf,look
%O A218339 1,2
%A A218339 _Alois P. Heinz_, Oct 26 2012