Vô, Pseudonym: Vô Conjecture 15 Series 8(VC15S8)

We investigate the Diophantine condition$$a+b=c, \quad \gcd(a,b,c)=1,$$together with the arithmetic constraint$$\Omega(\sigma(\varphi(abc)))=\Omega(abc),$$where $\varphi$ denotes Euler’s totient function, $\sigma$ the sum-of-divisors function, and $\Omega(n)$ the total number of prime factors of $n$, counted with multiplicity.