For prime $p$ let $E\_p[\dots]$ and $P\_p[\dots]$ be the external and polynomial $\mathbb{Z}\_p$--algebras.


It is known that for $n\geqslant 1$ and odd $p$ where is an isomorphism
of primitively generated Hopf algebras
$H\_\*(\Omega^2S^{2n+1};\mathbb{Z}\_p)=E\_p[x\_0,x\_1,x\_2,\dots]\otimes P\_p[y\_1,y\_2,\dots]$
where $\deg(x\_i)=2p^in-1$ and $\deg(y\_i)=2p^in-2$.


For $p=2$ where is an isomorphism
of primitively generated Hopf algebras
$H\_\*(\Omega^2S^{2n+1};\mathbb{Z}\_2)=P\_2[z\_0,z\_1,z\_2,\dots]$
where $\deg(z\_i)=2^{i+1}n-1$.


$\mathbf{Question:}$ How does a similar description of $H\_\*(\Omega^3S^{2n+1};\mathbb{Z}\_p)$ looks like for $n\geqslant 2$ ?



Homology of iterated loop spaces on odd--dimensional spheres
For the case where $n=dp$, the calculation of $H\_\*(\Omega^3S^{2n+1};\mathbb{Z}/p)$ is reviewed in Section 4.2 of the paper [The triple loop space approach to the telescope conjecture](https://people.math.rochester.edu/faculty/doug/mypapers/obit.pdf) by Mahowald, Ravenel and Shick. I don't think that the assumption that $n=dp$ makes any real difference, although it will take a little work to straighten out all the indices for the general case. All the real work is done in the book [Homology of iterated loop spaces](http://www.math.uchicago.edu/%7Emay/BOOKS/homo_iter.pdf) by Cohen, Lada and May; the paper of Mahowald, Ravenel and Shick just spells out the simplifications that occur in the case at hand.


