Here is a slightly more fleshed out version of my comment. Let $K(1)$ be the first Morava $K$-theory. When $p$ is odd one can calculate the homotopy groups of the $K(1)$-localised sphere spectrum to be
$$
\pi_nL_{K(1)}\mathbb{S} = \begin{cases}
\mathbb{Z}_p, &n=0,1\\
\mathbb{Z}/p^{\nu_p(t')+1} & n=2(p-1)t'-1, t' \in \mathbb{Z}.
\end{cases}
$$
Here $\nu_p(x)$ is the $p$-adic valuation of $x$.

Following Adams define a function $m(l)$ by
$$
\nu_p(m(l)) = \begin{cases}
0 & l \not \equiv 0 \mod (2(p-1)) \\
1+ \nu_p(l) & l \equiv 0 \mod (2(p-1)).
\end{cases}
$$
Adams shows (following Milnor and Kervaire) that $m(2s)$ is the denominator of $\beta_{2s}/4s$, where $\beta_s$ is the $s$-th Bernoulli number, and the fraction is expressed in the lowest possible form.

Using standard properties of $\nu_p(x)$ there is an equivalence $\nu_p(t')+1 = \nu_p((n+1)/2)+1$. Since $(n+1)/2 \equiv 0 \mod (2(p-1))$ we see
$$
\nu_p(m\left(2\cdot \frac{n+1}{4}\right)) = \nu_p((n+1)/2)+1
$$
and so the order of $\pi_nL_{K(1)}S^0$ is the denominator of $\beta_{(n+1)/2}/(n+1)$.

**Edit:** Let me try and say something about the image of $J$ then. This is a homomoprhism $J:\pi_nSO \to \pi_n\mathbb{S}$. When $n=4k-1$ the order of the image of $J$ is cyclic of order the denominator of $\beta_{2k}/4k$. Let $\text{Im}(J_n)_p$ denote the image of the composite $\pi_nSO \to \pi_n\mathbb{S} \to \pi_n \mathbb{S}_{(p)}$. I believe this is meant to be isomorphic to $\pi_nL_{K(1)}\mathbb{S}$ for $n>1$.