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Let $F$ be a (nontrivial) topological space that satisfies the following condition: $\pi_n(F)$ has a trivial action of $\pi_1(F)$ for $n>0$. 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:

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?)

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:

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?)