I can deduce the following simple proposition, the definitions for $\sigma(x)$ the sum of divisors functions and $\varphi(x)$ the Euler totient function are assumed. After I present a conjecture that I've tested for the square $1000\times 1000$. (Please add feedback in comments if you think that the question can be improved before downvote it.) The sequence from the OEIS related to the post is *A023194*.

**Claim.** *Let $B\geq1$ and $C\geq 1$ be positive integers with $B$ a prime numbers satisfying $$\sigma(C)=B,\tag{1}\label{1}$$
then $\varphi(B)$ has the factorization
$$\varphi(B)=\operatorname{rad}(C)\cdot (B-C),\tag{2}\label{2}$$
where $\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$ denotes the product of distinct primes dividing an integer $n> 1$.*

One also can deduce under the same assumptions of **Claim** the following identities $\varphi(C)=C\left(1-\frac{1}{\operatorname{rad}(C)}\right)$, $\frac{\varphi(C)}{C}=\frac{C-1}{B-1}$ and $\sigma(C^{1+\lambda})=\frac{C^{\lambda}-1}{\operatorname{rad}(C)-1}+BC^{\lambda}$, for $\lambda\geq 1$ integer.

**Conjecture.** *If \eqref{1} and \eqref{2} hold for some integers $B$ and $C$ strictly greater than $1$, then $B$ is a prime number.*

Question.I would like to know if it is possible to prove previous conjecture, orwhat work can be doneabout it.

I was inspired in an edited question (on Mathematics Stack Exchange, now closed with identificator **4423186**) that I'm going to delete from my profile on MSE.

**As application (that I evoke)** I tried, and I invite to it if you think that it is interesting, to combine these identities for an odd perfect number of the form (for example) $BCM^2$, for pairwise coprime integers $B$, $C$ and $M$ under the previous assumptions.

`\label`

+`\eqref`

works here. Further, there is no need to end and begin italics around math mode; for example, "If \eqref{1} and \eqref{2} hold for some integers $B$ and $C$"`*If \eqref{1} and \eqref{2} hold for some integers $B$ and $C$*`

works just fine, and you need not do "If \eqref{1} and \eqref{2} hold for some integers$B$and$C$"`*If \eqref{1} and \eqref{2} hold for some integers* $B$ *and* $C$`

. I have edited accordingly. $\endgroup$that isn't relatedto our post, is that at my home I define the arithmetic function $<x,y>=\varphi(y)-\operatorname{rad}(x)\cdot (y-x)$ for positive integers $x,y\geq 1$ and I call itthe refraction bracket. This arithmetic function satisfies the properties a) $<x,x>=\varphi(x)$ $\forall x\geq 1$ integer and b) for a given integer $x$ and a prime number $y$ satisfying $\sigma(x)=y$ then $<x,y>=0$. While that at my home I call thechiral bracketthe arithmetic funciton $[x,y]$ defined in our question of this MathOverflow with identificator419954$\endgroup$`\langle x, y\rangle`

, not $<x, y>$`<x, y>`

. Notice especially the difference in spacing between, e.g., $2\langle x, y\rangle$`2\langle x, y\rangle`

and $2<x, y>$`2<x, y>`

. $\endgroup$1more comment