TITLE: 3-primary part of $\pi_\bullet S^3$
QUESTION [8 upvotes]: I am looking for the most updated calculation of $\pi_\bullet S^3$ at the prime 3 (and also at any other odd primes).
The motivation of this question is just curiousity: since $\pi_\bullet S^3$ has exponent $p$ at any odd prime, these homotopy groups are completely determined by the number of generators and I was wondering how this sequence of numbers looks like.
On Toda book the computation goes to the 19+3 group. I have also found computation up to 64, but only at the prime 2.

REPLY [3 votes]: Gregory Arone told me that there is an unstable Adams spectral sequence, which I was not aware of. I found the $E_2$ page for the 3-primary part of $S^3$.

I think it doesn't contain the differentials. Without differentials, one can be sure only up to the 19-stem, as in Toda's Composition Methods in Homotopy Groups of Spheres, but at least it gives a partial answer up to the 81-stem, until one will find a copy of Toda's Unstable 3-Primary Homotopy Groups of Spheres.
The number of copies of $\mathbb{Z}_3$ in $\pi_{3+*}S^3_{(3)}$ up to the 19-stem is 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1.