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May 1, 2017 at 16:02 vote accept Jens Reinhold
May 1, 2017 at 6:19 comment added მამუკა ჯიბლაძე The answer already given below, a next natural question is whether it would be enough to require the same conditions but, instead of all primes $>c$, only for some infinite set of primes.
May 1, 2017 at 0:56 comment added Aaron Meyerowitz A minor comment, but , since only the gaps count, you can adjust the indexing , weaken to non-negative, and assume $a_0=0$ then for $k > c$ we have that $a_k$ has all the prime factors of $k$ with the possible inclusion or exclusion of primes less than $c+1$
Apr 30, 2017 at 23:39 answer added Julian Rosen timeline score: 22
Apr 30, 2017 at 23:14 history asked Jens Reinhold CC BY-SA 3.0