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We denote as $N$ an odd perfect number, and $d\mid N$ one of its divisors. We denote the Carmichael function as $\lambda(x)$, Wikipedia has the article Carmichael function dedicated to this number theoretic function.
Question. I would like to know if on assumption that $N$ is an odd perfect number we can to prove or refute that there exists a divisor $d\mid N$ with $$1<d<N\tag{1}$$ such that $$\lambda\left(\frac{N}{d}\right)\mid \sigma\left(\frac{N}{d}\right)\tag{2}$$ holds, where $\sigma(x)$ denotes the sum of divisor function $\sum_{1\leq d\mid x}d$. Many thanks.
I've known the definition of Carmichael function from page 21 of the book [1] and the propositions and examples showed in pages 144-146 (below CMS means Canadian Mathematical Society). I think that this question isn't in the literature. I don't know if it is possible to deduce some proposition about my question.
References:
[1] Michal Krizek, Florian Luca, and Lawrence Somer, 17 Lectures on Fermat Numbers, CMS Books in Mathematics, Springer (2001).
