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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>.
1
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
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: A109301 A107573 A376569 * A081308 A070210 A100198
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Aug 11 2024
STATUS
approved