Let $d(n)$ and $d^{ \ast}(n)$ be the numbers of divisors and the numbers of unitary divisors of the integer $n\geq1$. In this paper, we prove that \[ \underset{ n\in\mathcal{ B}}{ \underset{ n\leq x}{ \sum}}\frac{ d(n)}{ d^{ \ast}% (n)}=\frac{ 16\pi% %TCIMACRO{ \U{ b2}}% %BeginExpansion { { }^2}% %EndExpansion }{ 123}\underset{ p}{ \prod}(1-\frac{ 1}{ 2p% %TCIMACRO{ \U{ b2}}% %BeginExpansion { { }^2}% %EndExpansion }+\frac{ 1}{ 2p^{ 3}})x+\mathcal{ O}\left( x^{ \frac{ \ln8}{ \ln10}+\varepsilon }\right) ,~\left( x\geqslant1,~\varepsilon>0\right) , \] where $\mathcal{ B}$ is the set which contains any integer that is not a multiple of $5,$ but some permutations of its digits is a multiple of $5.$
Categories: math.NT