A375200 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>.

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, 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, 2, 2, 2, 2, 4, 4, 8, 32

Offset

1,2

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..73.

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
 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, 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;
 2, 2, 2,  2,  4,  4,  8, 32;
...

Mathematica

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}]

Crossrefs

Cf. A375312.

Sequence in context: A128886 A031883 A366261 * A086152 A194577 A334970

Adjacent sequences: A375196 A375197 A375198 * A375219 A375220 A375221

Keyword

nonn,tabf,new

Author

Geoffrey Critzer, Aug 11 2024

Status

approved