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Are there infinitely many $n$ such that, for all $k\geq 1$, \[ \omega(n+k) \ll k,\] where the implied constant depends only on $n$? (Here $\omega(n)$ is the number of distinct prime divisors of $n$.)
Related to [69]. Erdős and Graham [ErGr80] write 'we just know too little about sieves to be able to handle such a question ("we" here means not just us but the collective wisdom (?) of our poor struggling human race).'

See also [679].