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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.

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    $\begingroup$ Toda's memoir Unstable 3-Primary Homotopy Groups of Spheres as discussed here is state of the art. Good luck tracking it down. $\endgroup$
    – Tyrone
    Commented Oct 22, 2020 at 11:52
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    $\begingroup$ @Tyrone Oh, well, I see.. it seems a problem to find it out! Thank you for your help $\endgroup$
    – CNS709
    Commented Oct 22, 2020 at 21:04

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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 $S^3$.

enter image description here

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.

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