6. Let $[x]$ denote the greatest integer not exceeding the real number $x$,
$$
a_{k}=\left[\frac{2014}{k}\right](k=1,2, \cdots, 100) \text {. }
$$

Then, among these 100 integers, the number of distinct integers is
6.69.

When $01$, $[a+b]>[a]$.
$$
\begin{array}{l}
\text { Also } \frac{2014}{k}-\frac{2014}{k+1}=\frac{2014}{k(k+1)} \\
\Rightarrow \frac{2014}{k}=\frac{2014}{k+1}+\frac{2014}{k(k+1)} .
\end{array}
$$

When $k \leqslant 44$, $\frac{2014}{k(k+1)}>1$;
When $k \geqslant 45$, $\frac{2014}{k(k+1)}<1$.
Therefore, when $k \leqslant 44$, $a_{1}, a_{2}, \cdots, a_{44}$ are all different,
$$
a_{44}=45, a_{45}=45, \cdots, a_{99}=20, a_{100}=20 \text {; }
$$

When $k \geqslant 45$, $a_{k}=a_{k+1}$ or $a_{k}=a_{k+1}+1$.
Thus, $a_{45}, a_{46}, \cdots, a_{100}$ take all the 25 integers from $20 \sim 44$.

Therefore, $a_{1}, a_{2}, \cdots, a_{100}$ contain a total of $44+25=69$ different integers.