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 ?
  • $\begingroup$ If there is an example for $k=7$ it must be $>10^{10}$. $\endgroup$ – Julián Aguirre Jun 3 '15 at 15:08

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