# About the number of τ-numbers relative to polynomials with integer coefficients

Keywords:
tau-numbers, polynomials with integer coefficients, number of divisors, anti-tau-numbers, tau-numbers relative to polynomials, generators of a tau-number

### Abstract

We show that for all polynomials *Q*(*x*) with integer coefficients, that satisfy the extra condition *|Q*(0) *· Q*(1) *|* ≠ 1, there are infinitely many positive integers *n *such that *n *is a *τ*-number relative to the polynomial *Q*(*x*). We also find some examples of polynomials *Q*(*x*) for which 1 is the only *τ*-number relative to the polynomial *Q*(*x*) and some examples of polynomials *Q*(*x*) with |*Q*(0) *· Q*(1)*|*= 1, which have infinitely many positive integers *n *such that *n *is a *τ*-number relative to the polynomial *Q*(*x*). In addition, we prove one result about the generators of a *τ*-number.

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