Unfortunately, the formula is still far from the true one. Please check the result in some program.

Thank you very much for answer! Could you please double check your final formula? As for my conjectured formula, I tested it separately for the first few $p$ without counterexamples. I would be very grateful if you could give me a working closed form for $a(n)$.

@მამუკაჯიბლაძე, please double check your result.

@მამუკაჯიბლაძე, it looks like we can permute $a(2n)$ with A059894 and it gives no changes. So A035928 is the double of its fixed points (A290254).