I prefer the term "bounded morphism" over "p-morphism", and " bounded morphism" is the more modern term, but I feel compelled to use "p-morphism" anyway because it happens to be the norm in the subfield my thesis is in.

I came across this nice integerological fact:

Define \(\operatorname{ord}_p(n)\) to be the power of \(p\) in the prime factorisation of \(n\).

Then

\[ \sum_{k=1}^n\gcd(n,k) = \prod_{p|n} \left(1+\left(1-\frac{1}{p}\right) \cdot \operatorname{ord}_p(n)\right) \]

I don't know why you'd want to do this, and all the references I've found to it talk about it as a special case of some more general identities, but for some reason it appealed to me.

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