Vô Conjecture 15 series 9(VC15S9)

We investigate the Diophantine condition d(\sigma(\varphi(abc)))=k\mathrm{\thinsp} d(abc),\bigm where d(n)denotes the divisor-counting function, \sigma(n)denotes the sum-of-divisors function, and \varphi(n)denotes Euler’s totient function. The variables a,b,care positive integers satisfying a+b=c,\gcd\funcapply(a,b,c)=1.\bigm Computational evidence suggests that for each fixed integer k\in{1,2,3,4,5},\bigm there exist infinitely many primitive triples abcsatisfying the equation. We formulate this statement as Vc15s9 and provide several explicit examples.