Complex bundles on $S^{2n+1}$

Question

We know that the complex $K$ theory of spheres are 2 periodic. On the other hand every complex bundle on $S^{1}$ is trivial. So $K(S^{2n+1})=0$. So this is a motivation to ask: Is there a non trivial complex bundle on $S^{2n+1}$ for some $n$?

Answer 1

Qiaochu Yuan:

Bott (I think?) proved that $\pi_{2n}(U(n)) \cong [S^{2n-1}, BU(n)] \cong \mathbb{Z}_{n!}$, so the odd spheres $S^{2n-1}$ have nontrivial $n$-dimensional complex vector bundles on them for all $n \ge 2$.