For any positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
let
![d(n)](/media/m/8/b/5/8b5ba2b86903af1640ec9f08b90773b6.png)
denote the number of positive divisors of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
. Do there exist positive integers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
, such that
![d(a)=d(b)](/media/m/e/9/6/e96104e8ea1412a5b5e8e205b5664b23.png)
and
![d(a^2 ) = d(b^2 )](/media/m/c/8/7/c87a2862557f7cd9324a6fcad647e5da.png)
, but
![d(a^3 ) \ne d(b^3 )](/media/m/c/1/4/c14c71bd5b6bae994cc8db134203e015.png)
?
%V0
For any positive integer $n$ let $d(n)$ denote the number of positive divisors of $n$. Do there exist positive integers $a$ and $b$, such that $d(a)=d(b)$ and $d(a^2 ) = d(b^2 )$, but $d(a^3 ) \ne d(b^3 )$ ?