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.