Consider the $k$-sphere (I'm particularly interested in $k=3$. For each positive integer $n$ let $P_{n}$ be the set of primes that divide the order of $\pi_{i}S^{k}$ for some $i\leq n$.

Does the cardinality of $P_{n}$ tend to infinity with $n$?

Consider the $k$-sphere (I'm particularly interested in $k=3$). For each positive integer $n$ let $P_{n}$ be the set of primes that divide the order of $\pi_{i}S^{k}$ for some $i\leq n$.

Does the cardinality of $P_{n}$ tend to infinity with $n$?