Let $F$ be a (nontrivial) topological space that satisfies the following conditions: 1) $\pi\_n(F)$ has a trivial action of $\pi\_1(F)$ for $n>0$ and 2) its homology groups are finitely generated. Then $\pi\_n(F)$ is finitely generated for $n>0$; it therefore makes sense to consider the following sum:
$$I(F)=\sum\_{q=1}\frac{\operatorname{Rank}(\pi\_q(F))}{q}.$$
I have two questions:



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> When does $I(F)$ converge? When is $I(F)$ a natural number? (Like is there a (necessary and) sufficient condition for $I(F)$ to converge and/or be a natural number?)
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Convergence of a sum with the ranks of homotopy groups
Suppose $F$ is a simply connected finite CW complex. Then it's known that exactly one of the following two things is true:


* $F$ is *rationally elliptic*: its rational homotopy groups are finitely generated. In this case your sum clearly converges because it has finitely many terms.
* $F$ is *rationally hyperbolic*: the ranks of its rational homotopy groups grow, on average, exponentially. In this case your sum clearly diverges because its terms don't go to zero.


For a simple example of the second case you can take $F = S^2 \vee S^2$. 


