  In this note, we prove by using T. Estermann's and S. Dimitrov's arguments
with an elementary inequality that there are infinitely many $n$ for which all
of the numbers $n^2+1,n^2+2$ and $n^2+3$ are squarefree. We also improve the
error term slightly in the case of two consecutive numbers of the same form, so
that we are able to prove the following asymptotic formula. \begin{align*}
\sum_{n\le
X}\mu^2(n^2+1)\mu^2(n^2+2)\mu^2(n^2+3)\sim\dfrac{7}{18}\prod_{p>3}\left(1-\dfrac{3+\left(\frac{-1}{p}\right)+\left(\frac{-2}{p}\right)+\left(\frac{-3}{p}\right)}{p^2}\right)X.
\end{align*}
