  The main topic of this contribution is the problem of counting square-free
numbers not exceeding $n$. Before this work we were able to do it in time
(Comparing to the Big-O notation, Soft-O ($\softO$) ignores logarithmic
factors) $\softO(\sqrt{n})$. Here, the algorithm with time complexity
$\softO(n^{2/5})$ and with memory complexity $\softO(n^{1/5})$ is presented.
Additionally, a parallel version is shown, which achieves full scalability.
  As of now the highest computed value was for $n=10^{17}$. Using our
implementation we were able to calculate the value for $n=10^{36}$ on a
cluster.
