In this post I add two equations involving the sum of divisors $\sigma(n)$ and the Euler's totient function, denoted in this post as $\varphi(n)$, and after I ask about a conjecture involving these. The first paragraph, after this introductory paragraph, is thus the context and I refer it to the literature.
In the nice [1] Iannucci presents an equation involving the Euler's totient function and the sum of divisors, that is the background of next claim.
Claim Let $n>2$ an integer satisfying Iannucci's equation then $n$ satisfies $$\varphi(n)+\sigma\left(2^{\varphi(n)-1}n\right)=\left(2^{\varphi(n)}-1\right)n+\varphi \left(2^{\varphi(n)+1}n\right).\tag{1}$$
Skech of the proof. Invoke Iannucci's Theorem 2 from [1] exploiting the fact that the Euler's totient function and the sum of divisors function are multiplicative. $\square$
To create the mentioned conjectures we make the substitution $N=2^{\varphi(n)}-1$ from $(1)$ the equation $(2)$, respectively the substitution $R=2^{\varphi(n)-1}$ to get $(3)$.
Conjecture. Let $N$ a positive integer such that $2\mid (N+1)$ (that is $N$ is an odd integer) and the equation $$\varphi(n)+\sigma\left(\left(\frac{N+1}{2}\right)n\right)=Nn+\varphi(2(N+1)n)\tag{2}$$ holds for some positive integer $n$, then $n$ is an even integer.
Similarly, let $1\leq R$ and $1\leq n$ positive integers such that satisfy $$\varphi(n)+\sigma(Rn)=(2R-1)n+\varphi(4Rn),\tag{3}$$ then $n$ is an even integer.
Question. I've tested the conjectures for positive integers up to $\leq 7000$. What work can be done about the veracity of previous conjectures, I say the statements of my Conjecture? Many thanks.
References:
[1] Douglas E. Iannucci, On the Equation $\sigma(n)=n+\phi(n)$, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.
