Solutions to σ(n+1) = k σ(n) have zero natural density with counting function A_k(x) ≪_k x / sqrt(log log log x), and are conditionally infinite for k=2.
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A generalization of the Erd\H{o}s-Sierpi\'nski conjecture
Solutions to σ(n+1) = k σ(n) have zero natural density with counting function A_k(x) ≪_k x / sqrt(log log log x), and are conditionally infinite for k=2.