We denote the Euler's totient as $\varphi(x)$, the Dedekind psi function as $\psi(x)$ and the sum of divisors function as $\sigma(x)$. Are well-known arithmetic functions, see the corresponding articles from Wikipedia or search these in this website Mathematics Stack Exchange. I would like to get idea about the veracity of the following conjecture (I have some computational evidence of it).
For integers $k> 1$ we denote the composition of functions $f^{k)}(n)=f^{k-1)}(f(n))$ with the definition $f^{1)}(n):=f(n)$.
Conjecture 1. Let $1\leq y<x$ be integers such that $2x=27(y-1)$, and satisfy the equations $$\frac{\psi^{2)}(x)}{\psi^{2)}(y)}=\left(\frac{\varphi^{2)}(x)}{\varphi^{2)}(y)}\right)^3\tag{1}$$ and $$\sigma(x)=k!\tag{2}$$ for some integer. Then $x=1836$ and $y=137$.
Computational evidence. The conjecture is true for $1\leq k\leq 10$ and $1\leq y<x\leq 3000$ (I agree that this is poor, but with the means at my disposal I cannot improve this).
Question. Is previous conjecture true? What work can be done about it? Many thanks.
I don't know if it is possible to find quickly a counterexample for Conjecture 1. In case that you're able to find quicly counterexample, please can you find a counterexample or provide what work can be done to discuss the veracity of Conjecture 2? (What I evoke is the situation in which you can to find a counterexample and the post in this case could to seem poor because my computational evidence is poor, in this case please consider to find counterexamples for Conjecture 2 or show some what work can be done to discuss the veracity of Conjecture 2). I hope you forgive me, in fact Conjecture 2 seems very nice.
My attempts make computations and to study cases, with the equations that I've deduce in nexts claims.
Claim 1. We've that the following statements are true: a) the equation $2x=27(y-1)$ implies that $y>1$ is an odd integer; b) we've that $27\mid x$ and $x$ can be represented as $x=2^\alpha 3^\beta m$ with $y-1=2^{\alpha+1} 3^{\beta-3} m$ for some integers $\alpha\geq 0$, $\beta\geq 3$ and $m\geq 1$ satisfying $\gcd(2,m)=\gcd(3,m)=1$, and c) the equation $(2^{\alpha+1}-1)(3^{\beta+1}-1)\sigma(m)/2=k!$ holds for some integer $k\geq 1$.
Now we use the notation $p^a||n$ for prime povers $p^a$ dividing an integer $n$ with $p^{a+1}\nmid n$.
Claim 2. Under the additional conditions that $x$ is even and $3^3||x$ (therefore $\beta=3$) we've that the following equations hold $\varphi(x)=2^{\alpha} 3^{\beta-1} \varphi(m)$,$\psi(x)=2^{\alpha+1} 3^{\beta} \psi(m)$,$\psi(y)=\psi(2^{\alpha+1} 3^{\beta-3} m+1)$, $\varphi(y-1)=2^\alpha\varphi(m)$ and $\psi(y-1)=2^{\alpha}\cdot 3 \psi(m)$.
We've the same computational evidence for next conjecture.
Conjecture 2. Let $1\leq y<x$ be integers such that $2x=27(y-1)$ with $x+y$ and $x-y$ are both prime numbers, and $x$ satisfying the equation $$\sigma(x)=k!$$ for some integer $k\geq 1$, then $x=1836$ and $y=137$.
References:
[1] Was edited an excellent answer for my recent post On a conjecture involving multiplicative functions and the integers $1836$ and $136$ on this Mathematics Stack Exchange.
[2] The sequence A245015 from The On-Line Encyclopedia of Integer Sequences is related to our problem.
I edit this post with numerical coincidences for the integers related to Conjecture 1 if it can be inspiring/help together the work that I show in the body of the post. I hope that there are no typos. On the other hand I refer that the person in comments (this comment) got counterexamples for Conjecture 2, and this user is invited to post the calculations as an answer.
Claim. For $k=7$ the integers $x=1836$ and $y=137$ satisfy
A) the equation $$(\varphi(x)+y)^2+4\sigma(x)=\left(k+\frac{\sigma(x)}{k}\right)^2$$ as a consequence of $\varphi(x)=\frac{\sigma(x)}{k}-(y+k)$, where the square is $(k+\sigma(x)/k)^2=727^2$. Addtionally $\varphi(x)=576$ and $576$ can be expressed as $576=4(y+k)$· Also we have $\sigma(2x)=2\sigma(x)+(k-1)!$ and therefore the equation $$\frac{\sigma(2x)}{2k+1}=\frac{\sigma(k)}{k}$$ holds.
B) we can represent the prime numbers $x+y=x+(k-1)!-\varphi(x)-k$ and $x-y=x+\varphi(x)-(k-1)!+k$ with $y$ expressed as $y=(k-1)!-\varphi(x)-k$, therefore $5y=(k-1)!-5k$. Additionally $4y-1$ and $6y-1$ are also prime numbers, thus we get the following representations for $7!$ $$5040=2(x+y)+2(4y-1)$$ and $$5040=2(x-y)+2(6y-1).$$
C) for $m=17$ our integers satisfy $$\psi(x)-2x=2\cdot\frac{x}{m}$$ and $y=1+8m$, thus $\psi(x)-2x=(m-1)\frac{x}{y-1}$ (here as tangential detail $\psi(x)=3888$).