jmc

For any prime number, the sum of its proper divisors is equal to $1$, so a prime number cannot be an abundant number. Therefore, it suffices to check the smallest composite numbers that are not divisible by $6$. We find that:

$\bullet$ for $4$, $1+2<4$, $\bullet$ for $8$, $1+2+4<8$, $\bullet$ for $9$, $1+3<9$, $\bullet$ for $10$, $1+2+5<10$, $\bullet$ for $14$, $1+2+7<14$, $\bullet$ for $15$, $1+3+5<15$, $\bullet$ for $16$, $1+2+4+8<16$, $\bullet$ for $20$, $1+2+4+5+10=22>20$.

Thus, the answer is $\boxed{20}$.