8. For positive integers $n \geqslant 2$, let $\alpha(n)$ denote the sum of all positive divisors of $n$ (including 1 and $n$). Define a positive integer $n$ as a "good number" if $(n, \alpha(n))=1$. How many consecutive good numbers can there be at most?
8. The maximum is 5.

On one hand, the five numbers 575, 576, 577, 578, 579 are all good numbers.
On the other hand, if a good number \( n \) is even, let
\[
n = 2^{\alpha} p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{s}^{\alpha_{s}},
\]
where \( p_{1}, p_{2}, \cdots, p_{s} \) are distinct odd primes, and \( \alpha, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{s} \in \mathbf{Z}_{+} \). Then
\[
\begin{array}{l}
\alpha(n) \\
= \left(2^{\alpha+1} - 1\right) \prod_{i=1}^{s} \left(1 + p_{i} + p_{i}^{2} + \cdots + p_{i}^{\alpha_{i}}\right).
\end{array}
\]
If \( \alpha_{i} \) is odd, then \( 1 + p_{i} + p_{i}^{2} + \cdots + p_{i}^{\alpha_{i}} \) is even, which contradicts \( (n, \alpha(n)) = 1 \).
Thus, \( \alpha_{i} (i=1,2, \cdots, s) \) must be even.
Therefore, \( n = x^{2} \) or \( 2 x^{2} \left(x \in \mathbf{Z}_{+}\right) \), meaning if \( n \) is even and good, then \( n \) is a perfect square or twice a perfect square.

If there are six consecutive good numbers, then there must be at least three consecutive even numbers that are good. Let these be \( 2(n-1), 2n, 2(n+1) \).
(1) If at least two of these three consecutive even numbers are perfect squares, since one of the adjacent even numbers is congruent to 2 modulo 4 and cannot both be perfect squares, the two perfect squares must be \( 2(n-1) \) and \( 2(n+1) \), with their difference being 4. Hence, there exist positive integers \( x, y \) such that \( y^{2} - x^{2} = 4 \), i.e.,
\[
(y - x)(y + x) = 4,
\]
which is impossible.
(2) If at least two of these three consecutive even numbers are twice perfect squares, then there exist positive integers \( x, y \) such that \( 2 y^{2} - 2 x^{2} = 1 \) or 2, which is also impossible.
In summary, it is impossible to find three consecutive even numbers that are all good.
Therefore, there are no six consecutive good numbers.