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S05 On the monotonicity of the divisor function.

05a. [Saint-Petersburg 1998] Let $d(n)$ denote the number of positive divisors of the number $n$. Prove that the sequence $d(n^2+1)$ does not become strictly monotonic from some point onwards.

05b. PEN J11 Prove that $d((n^2+1)^2)$ does not become monotonic from any given point onwards.

Here is the official solution file: PEN05S
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