If $n$ is odd then $S^{n-1}$ doesn't admit a nowhere-vanishing vector field, and if $n$ is even then there does exist one (Hairy Ball Theorem). One can then ask, on $S^{n-1}$, what is the maximum number of linearly independent vector fields? Denoting this number by $k(n)$ and rewriting $n=2^{4a+b}(2s+1)$, Adams computes $k(n)=2^b+8a-1$.

In particular, on the $(4n+1)$-spheres, there is only one nowhere-vanishing vector field up to linear-independence. We can see this on the circle, where the vector fields generated by (counter)clockwise rotation are the same up to an appropriate scalar-multiplication.

Is there a nice differential/topological reasoning behind this? Or stated another way, is there a down-to-earth way to deduce this result on $S^{4n+1}$?

Could there possibly be an analogous index theorem going on here, in the same way that the Poincare-Hopf theorem provides us the Hairy Ball result?