Multidegrees of Tame automorphisms with one prime number

Let $3leq d_1leq d_2leq d_3$ be integers. We show the following results: (1) If $d_2$ is a prime number and $frac{d_1}{gcd(d_1,d_3)} eq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or $d_3in d_1mathbb{N}+d_2mathbb{N}$; (2) If $d_3$ is a prime number and $gcd(d_1,d_2)=1$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_3in d_1mathbb{N}+d_2mathbb{N}$. We also relate this investigation with a conjecture of Drensky and Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials, and we give a counter-example to this conjecture.