Shortly, the idea is to find such series which admits "lazy" calculation:
instead of computing all the terms, it would be enough
to calculate its even terms (the case with $n=2$), and then multiply result by $2$;
or calculate third of its terms (the case with $n=3$), and then multiply result by $3$; etc.
It seems that we can construct such series for finite set of values $n$ (say, for $n\in\{1,2,4,8\}$ or $n \in \{1,2,3,4\}$).
So, my Question is: does there exist a series $$\sum_{k=0}^{\infty} u_{k}=1,$$ such that for each $n\in\mathbb{N}$: $$n\sum_{k=0}^{\infty} u_{nk}=1. \quad (?) \tag{1}$$
