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A212485 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(5) listed in ascending order.

%I A212485 #38 Oct 24 2022 11:09:48
%S A212485 1,2,4,3,6,8,12,24,31,62,124,13,16,26,39,48,52,78,104,156,208,312,624,
%T A212485 11,22,44,71,142,284,781,1562,3124,7,9,14,18,21,28,36,42,56,63,72,84,
%U A212485 93,126,168,186,217,248,252,279,372,434,504,558,651,744,868,1116
%N A212485 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(5) listed in ascending order.
%C A212485 The elements m of row n, are also solutions to the equation: multiplicative order of 5 mod m = n, with gcd(m,5) = 1, cf. A050977.
%D A212485 R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 557-560.
%H A212485 Alois P. Heinz, <a href="/A212485/b212485.txt">Rows n = 1..32, flattened</a>
%H A212485 V. I. Arnol'd, <a href="https://doi.org/10.1070/RM2003v058n04ABEH000641">Topology and statistics of formulas of arithmetics</a>, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28
%F A212485 T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (5^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,4}.
%F A212485 |M(n)| = Sum_{d|n} mu(n/d)*tau(5^d-1) = A059887.
%e A212485 Triangle T(n,k) begins:
%e A212485    1,  2,   4;
%e A212485    3,  6,   8, 12,  24;
%e A212485   31, 62, 124;
%e A212485   13, 16,  26, 39,  48,  52,  78,  104,  156, 208, 312, 624;
%e A212485   11, 22,  44, 71, 142, 284, 781, 1562, 3124;
%e A212485   ...
%p A212485 with(numtheory):
%p A212485 M:= proc(n) option remember;
%p A212485       `if`(n=1, {1, 2, 4}, divisors(5^n-1) minus U(n-1))
%p A212485     end:
%p A212485 U:= proc(n) option remember;
%p A212485       `if`(n=0, {}, M(n) union U(n-1))
%p A212485     end:
%p A212485 T:= n-> sort([M(n)[]])[]:
%p A212485 seq(T(n), n=1..8);
%t A212485 M[n_] := M[n] = If[n == 1, {1, 2, 4}, Divisors[5^n-1] ~Complement~ U[n-1]];
%t A212485 U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
%t A212485 T[n_] := Sort[M[n]]; Array[T, 8] // Flatten (* _Jean-François Alcover_, Jun 10 2018, from Maple *)
%Y A212485 Cf. A212906, A059912, A058944, A059499, A059886-A059892.
%Y A212485 Column k=3 of A212737.
%Y A212485 Column k=1 gives: A218357.
%K A212485 easy,nonn,look,tabf
%O A212485 1,2
%A A212485 _Boris Putievskiy_, Jun 02 2012