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Mathematics, Number Theory
Abstract:
The notion of block divisibility naturally leads one to introduce unitary
cyclotomic polynomials $\Phi_n^*(x)$. They can be written as certain products
of cyclotomic poynomials. We study the case where $n$ has two or three distinct
prime factors using numerical semigroups, respectively Bachman's
inclusion-exclusion polynomials. Given $m\ge 1$ we show that every integer
occurs as a coefficient of $\Phi^*_{mn}(x)$ for some $n\ge 1$. Here $n$ will
typically have many different prime factors. We also consider similar questions
for the polynomials $(x^n-1)/\Phi_n^*(x),$ the inverse unitary cyclotomic
polynomials.
