The second approximation conjecture for ν₂(S(m,5))

Let $x_1(m)$, $I_1$, $f_1(m)$, and $\operatorname{Err}_1(m,5)=\nu_2(S(m,5))-\nu_2(f_1(m))$ be as above. Define

$$\lambda_p(m)=\frac12\left(1-(-1)^{m\bmod p}\right),$$ $$m_3(m)=(m+2)\bmod3,$$ $$\alpha_m=\lambda_3(m+2)(1+\lambda_3(m))+\lambda_2(m+1)\lambda_3(m),$$

and

$$f_2(m)=\binom{2m_3}{m_3}\left\lfloor\frac{m+2}{3}\right\rfloor+208\lambda_3(m+1)+27\lambda_2(m)\lambda_3(m).$$

Set $I_2=\{x_2(m):m\geq0\}$, where

$$x_2(m)=109+107\left\lfloor\frac{4m+2}{3}\right\rfloor+85\left\lfloor\frac{4m+1}{3}\right\rfloor.$$

The second approximation conjecture.

$$\operatorname{Err}_1(x_1(m),5)=(-1)^{\alpha_m}\nu_2(f_2(m))$$

unless $m\in I_2$. This is an empirical improvement of the first approximation, and the supplied text gives no proof or resolution.