A375312 - OEIS

%I #11 Aug 12 2024 02:57:24

%S 2,2,3,2,3,3,2,3,9,2,3,3,9,2,3,3,3,9,2,3,3,9,9,2,3,3,3,9,9,2,3,3,3,3,

%T 9,9,2,3,3,3,3,9,27,2,3,3,3,3,3,9,27,2,3,3,3,3,3,3,9,27,2,3,3,3,3,3,9,

%U 9,27,2,3,3,3,3,3,3,9,9,27,2,3,3,3,3,3,3,3,9,9,27

%N Irregular triangular array read by rows.  The n-th row gives the elementary divisors of the group of units in the quotient ring F_3[x]/<x^n>.

%C A general formula for the isomorphism class of the group of units in any quotient ring of the polynomial ring F_p[x] (p prime) is given by Keith Kearnes in the Mathematics Stack Exchange link below.

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/1956919/multiplicative-group-modulo-polynomials">Multiplicative group modulo polynomials</a>

%H Karlee Westrem, <a href="https://conservancy.umn.edu/server/api/core/bitstreams/56a0c4f3-1a9f-4e3d-a10a-bde8bcfc9dba/content">Group of Units of Z_p[x] modulo f(x)</a>, Masters Thesis, University of Minnesota, 2020.

%e Triangle begins

%e  2;

%e  2, 3;

%e  2, 3, 3;

%e  2, 3, 9;

%e  2, 3, 3, 9;

%e  2, 3, 3, 3, 9;

%e  2, 3, 3, 9, 9;

%e  2, 3, 3, 3, 9, 9;

%e  2, 3, 3, 3, 3, 9,  9;

%e  2, 3, 3, 3, 3, 9, 27;

%e  2, 3, 3, 3, 3, 3,  9, 27;

%e ...

%t groupofunits3xn[e_] := Flatten[Prepenf[Table[Table[3^(i + 1), {Ceiling[e/3^i] - 2 Ceiling[e/3^(i + 1)] + Ceiling[e/3^(i + 2)]}], {i, 0, 10}], 2]];

%t Table[groupofunits3xn[n], {n, 1, 15}] // Grid

%Y Cf. A375200.

%K nonn,tabf

%O 1,1

%A _Geoffrey Critzer_, Aug 11 2024