A375200 - OEIS

%I #26 Jan 08 2025 11:39:48

%S 1,2,4,2,4,2,8,2,2,8,2,4,8,2,2,4,8,2,2,4,16,2,2,2,4,16,2,2,4,4,16,2,2,

%T 2,4,4,16,2,2,2,4,8,16,2,2,2,2,4,8,16,2,2,2,4,4,8,16,2,2,2,2,4,4,8,16,

%U 2,2,2,2,4,4,8,32

%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_2[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  1;

%e  2;

%e  4;

%e  2, 4;

%e  2, 8;

%e  2, 2, 8;

%e  2, 4, 8;

%e  2, 2, 4,  8;

%e  2, 2, 4, 16;

%e  2, 2, 2,  4, 16;

%e  2, 2, 4,  4, 16;

%e  2, 2, 2,  4,  4, 16;

%e  2, 2, 2,  4,  8, 16;

%e  2, 2, 2,  2,  4,  8, 16;

%e  2, 2, 2,  4,  4,  8, 16;

%e  2, 2, 2,  2,  4,  4,  8, 16;

%e  2, 2, 2,  2,  4,  4,  8, 32;

%e ...

%t groupofunits2xn[e_] := Flatten[Table[{Table[2^(i + 1), {(Ceiling[e/2^i] - 2 Ceiling[e/2^(i + 1)] + Ceiling[e/2^(i + 2)])}]}, {i, 0, 10}]]; Prepend[Drop[Table[groupofunits2xn[n], {n, 1, 16}], 1], {1}]

%Y Cf. A375312.

%K nonn,tabf

%O 1,2

%A _Geoffrey Critzer_, Aug 11 2024