I'm trying to think about the following types of problems and, in fact, if we could get results, they could be applied inside mathematics, even to unresolved problems.

I will not write any potential applications now, but will pass to the direct presentation of a problem.

I'm studying (though, no conclusions yet) the sum $$ sum_ {3 leq a_ {1,1}, a_ {1,2} <+ infty} frac {1} {a_ {1,1} a_ {1,2}} + sum_ {3 a_ {2,1}, a_ {2,2}, a_ {2,3} <+ infty} frac {1} {a_ {2,1} a_ {2,2} a_ {2,3 }} + … + sum_ {3 leq a_ {n, 1}, a_ {n, 2}, …, a_ {n, n + 1} <+ infty} frac {1} { a_ {n, 1} a_ {n, 2}, … a_ {n, n + 1}} + … $$.

or $ {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 zero density $ mathbb N $ and every term in every sequence is strange.

Does all this imply the convergence of this infinite sum of infinite sums?