3 research outputs found
The Frobenius Formula for
Given relative prime positive integers , the
Frobenius number is the largest integer not representable as a linear
combination of the 's with nonnegative integer coefficients. We find the
``Stable" property introduced for the square sequence naturally extends for . This gives a
parallel characterization of as a ``congruence class function" modulo
when is large enough. For orderly sequence , we
find good bound for . In particular we calculate for
, and
A Generalization of Mersenne and Thabit Numerical Semigroups
Let be relative prime positive integers with
. The Frobenius number is the largest integer not belonging
to the numerical semigroup generated by . The genus
is the number of positive integer elements that are not in . The Frobenius problem is to find and for a given
sequence . In this paper, we study the Frobenius problem of
and obtain formulas for and
when . Our formulas simplifies further for some special
cases, such as Mersenne and Thabit numerical semigroups. We obtain explicit
formulas for generalized Mersenne and Thabit numerical semigroups and some more
general numerical semigroups