I am trying to think about the following type of problems, and, actually, if we could get some results, they could be applied inside of mathematics, even to some unsolved problems.
I will not now write about potential applications, but proceed to a direct presentation of a problem.
i am studying (although, with no conclusions yet) the sum $$\sum_{3 \leq a_{1,1},a_{1,2}< + \infty} \frac {1}{a_{1,1}a_{1,2}}+\sum_{3 \leq a_{2,1},a_{2,2},a_{2,3}< + \infty} \frac {1}{a_{2,1}a_{2,2}a_{2,3}}+...+ \sum_{3 \leq a_{n,1},a_{n,2},...,a_{n,n+1}< + \infty} \frac {1}{a_{n,1}a_{n,2},...a_{n,n+1}}+...$$.
where $\{a_{1,1}a_{1,2}\}$ and $\{a_{2,1}a_{2,2}a_{2,3}\}$ and ... and $\{a_{n,1}a_{n,2},...a_{n,n+1}\}$ and... are all of density zero in $\mathbb N$ and every term of every sequence is odd.
Does all of this implies convergence of this infinite sum of infinite sums?