  In the paper we can prove that every integer can be written as the sum of two
integers, one perfect square and one squarefree. We also establish the
asympotic formula for the number of representations of an integer in this form.
The result is deeply related with the divisor function. In the course of our
study we get an independent result about it. Concretely we are able to deduce a
new upper bound for the divisor function valid for any integer and fully
explicit.
