Special $\omega(n)$-sequence

Question

Let $k$ be a natural number, $\omega(n)$ the number of distinct prime factors of $n$.

The object is to find a number $n$ with $\omega(n+j)=j+1$ for each $j$ with $0\le j\le k-1$. In other words, a sequence of $k$ numbers, such that the number of distinct prime factors increases from $1$ to $k$.

The least examples for some $k$ :

  k               n

  2               5
  3              64
  4            1867
  5          491851
  6        17681491

Questions :

• Does a number $n$ exists for each $k$ ?
• Is there a good estimate for the magnitude of the least example ?
• Can anyone extend the table ?