Growth of stable homotopy groups of spheres

There is work by Boedigheimer and Henn that bounds the size of unstable homotopy groups of spheres or rather of the number of $p$-local summands (i.e. the dimension after tensoring with $\mathbb{F}_p$). The bound is again exponential, namely $3^{q-n/2}$ for $\mathrm{dim}_{\mathbb{F}_p}\pi_q(S^n)\otimes \mathbb{F}_p$. There is a slight improvement in later work by Henn, but the bound is still exponential as I understand it.

Looking at the data, the growth of the stable homotopy groups seems to be less than exponential though. According to Isaksen's charts (with possible miscounts by myself) the sequence of the first few $k_n$ is:

1 1 3 0 0 1 4 2 3 1 3 0 0 2 6 2 4 4 4 3 2 2 8 2 2 2 3 1 0 1 8 4 5 5 5 1 2 3 9 7 5 5 3 3 7 4 10

Particularly big ones are $k_{15} = 6$, $k_{23} = 8$ and $k_{47} = 10$. The contribution of the image of $J$ is $5$, $4$ and $5$ respectively in these degrees. While the image of $J$ should dominate in low degrees, elements of higher Adams-Novikov filtration should become more and more dominant. All in all, the data does not really look like an exponentially growing sequence, but who knows with our limited knowledge?

Edit: I incorporated Allen Hatcher's corrections to my sequence of $k_n$.

Since posting the preprint Tim mentions, I found Iriye's 1987 paper `On the ranks of homotopy groups of a space'. Iriye says (Theorem 1 and Remark 2) that it is not hard to replace the $2^\frac{k}{p-1}$ in Tim's answer with a $3^\frac{k}{2p-3}$. This is slightly better than the bound I proved, and I suspect that the proof is cleaner (though Iriye doesn't give a proof, and I haven't checked it). I have updated the Arxiv submission to acknowledge Iriye. Of course, this is still an exponential bound.

A recent preprint of Boyde improves this bound, showing that

$$\log_p(\#\pi_{n+k}(S^n)_{(p)}) \leq c_p 2^{k/(p-1)}$$

where $c_p = \frac{1}{4}2^{1/(p-1)}$

Note that this bound depends on $k$ and not $n$, so it stabilizes to show that

$$\log_p(\#\pi^s_{k}(\mathbb S)_{(p)}) \leq c_p 2^{k/(p-1)}$$

In his introduction, Boyde mentions some earlier results of Henn which are better than the ones that Lennart mentions, in that they depend only on $k$ and so stabilize. The citation is to the same single-author paper of Henn that Lennart links to.

Of course, this is still an exponential bound.