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Let $\sigma(m)=\sum_{d\mid m}d$ the sum of divisors function, and $\varphi(m)$ the Euler's totient function, then it is possible to prove the following statements. And I would like to identify some related sequences about those. I don't know if these were in the literature.

Claim 1. If (there exists) $n$ is an odd perfect number satisfying $$\gcd(n,3)=\gcd(n,7)=1,$$ then our odd perfect number satisfies each of these equations $$\sigma(4\sigma(4n))=2\sigma(\sigma(3\sigma(n)))\tag{1}$$ and $$\sigma(\sigma(6\sigma(n)))=\sigma(4\sigma(4n)).\tag{2}$$

Question 1. Can you identify from OEIS the sequences of odd integers satisfying $(1)$ or $(2)$? Can you find an odd integer $n\geq 1$ satisfying $(1)$ or $(2)$? Can you find an odd integer $n\geq 1$ satisfying $(1)$ and $(2)$? If these sequences or equations are known from the literature answer this question as a reference request, and I try to find and read such literature. Many thanks.

Claim 2. If (there exists) $n$ is an odd perfect number satisfying $$\gcd(n,3)=1,$$ then our odd perfect number satisfies each of these equations $$\varphi(72n^2)=\varphi(2\sigma(2n)\sigma(\sigma(n)))\tag{3}$$ and $$\varphi(2\sigma(2n)\sigma(\sigma(n)))=24\varphi(n^2).\tag{4}$$

Question 2. Can you identify from OEIS the sequences of odd integers satisfying $(3)$ or $(4)$? Can you find an odd integer $n\geq 1$ satisfying $(3)$ or $(4)$? Can you find an odd integer $n\geq 1$ satisfying $(3)$ and $(4)$? If these sequences or equations are known from the literature answer this question as a reference request, and I try to find and read such literature. Many thanks.

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The sequence of integers satisfying (1) begins $$ 56,424,3566,5040,6000,6768,8240,8359,8949,11053, \dots . $$ It is not in the OEIS. It contains quite a few odd terms.

The sequence of integers satisfying (2) begins $$ 28,72,118,336,390,472,486,496,498,574,598,676,3343,3823,4363,\dots .$$ It also contains lots of odd terms. It is not in the OEIS.

$336$ is the only integer less than $10^6$ satisfying (3).

$147456$ is the only integer less than $10^6$ satisfying (4).

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    $\begingroup$ Many thanks, your are generous with your answer. I am going to wait more answers, but many thanks, it is a great help. $\endgroup$ – user243301 Jan 15 '18 at 6:22

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