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@ZachHunter Thanks! I was writing up my article, and I thought this might be difficult and thus not worth diverting my time. I really should have thought about it more.
@fedja Very nice. Out it goes, then. It was not the focus of my paper, but now I am interested: there is a small chance that $n_{k+1}/n_k > \lambda$ can be weakened to $n_{k+1}/n_k > 1 + o(1)$. I expect this gives a different answer.
