A375312 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>.

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, 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, 9, 27, 2, 3, 3, 3, 3, 3, 3, 9, 9, 27, 2, 3, 3, 3, 3, 3, 3, 3, 9, 9, 27

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..90.

Mathematics Stack Exchange, Multiplicative group modulo polynomials

Karlee Westrem, Group of Units of Z_p[x] modulo f(x), Masters Thesis, University of Minnesota, 2020.

Example

Triangle begins
 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, 9,  9;
 2, 3, 3, 3, 3, 9, 27;
 2, 3, 3, 3, 3, 3,  9, 27;
...

Mathematica

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

Crossrefs

Cf. A375200.

Sequence in context: A238459 A109301 A107573 * A081308 A070210 A100198

Adjacent sequences: A375307 A375308 A375311 * A375314 A375315 A375316

Keyword

nonn,tabf,new

Author

Geoffrey Critzer, Aug 11 2024

Status

approved