Let $a>1$ be an integer, and consider the infinite abelian group $$ V_a=\bigoplus_{j=1}^{\infty}\mathbb Z/{a^j\mathbb Z}. $$
Can anyone provide references to places where this (or related) groups arise in the literature? The groups $V_4$ and $V_8$ have recently cropped up in a problem I am working on.
As far as I know, this group does not naturally appear anywhere.
A closely related group is the "Prufer $a$-group" (in quotes because it is usually only defined for $a$ a prime), which is a certain quotient of this group. Its Pontryagin dual is the "$a$-adic integers" (again, in quotes because it is usually only defined for $a$ a prime), which is a certain subgroup of the corresponding infinite product. Both of these groups show up in various places.