Lather, rinse, repeat

$444_a = 44_b\\ \Rightarrow 4(111_a) = 4(11_b)\\ \Rightarrow 4(a^2 + a + 1) = 4(b + 1)\\ \Rightarrow a^2 + a = b$

Since the smallest possible value for $a$ is $5$ (otherwise you can't have the digit $4$ ), the smallest possible value for $b$ is $5^2 + 5 = 30$ .

This problem came out of noticing this interesting pattern:

$111_2 = 11_6$ (and $6 = 2 \cdot 3$ )

$222_3 = 22_{12}$ (and $12 = 3 \cdot 4$ )

$333_4 = 33_{20}$ (and $20 = 4 \cdot 5$ )

As above, this is because $111_a = a^2 + a + 1 = a(a+1) + 1$ , while $11_b = b + 1$ , so you can simply set $b = a(a+1)$ .

(And thinking about that pattern came from this comic .)