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
- 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 ?