Smallest Consecutive Even Numbers such that Added to Divisor Count are Equal

Theorem
$30$ is the smallest positive even integer $n$ with the property:

where:
 * $m \in \Z_{>0}$ is some positive integer
 * $\map \tau n$ is the $\tau$ function: the number of divisors of $n$.

In this case, where $n = 30$, we have that $m = 38$.

Proof
From Tau Function from Prime Decomposition, we have:
 * $\displaystyle \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$

where the prime decomposition of $n$ is:
 * $n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$