The solutions of the equation $$\frac{\phi(n)}{\phi(n-1)}=5$$ upto $n=10^8$ , where $\phi(n)$ denotes the totient function, are :
? for(n=2,10^8,if(eulerphi(n)/eulerphi(n-1)==5,print(n," ",factor(n-1)," ",fa
ctor(n))))
11242771 [2, 1; 3, 1; 5, 1; 7, 1; 11, 1; 31, 1; 157, 1] [1171, 1; 9601, 1]
18673201 [2, 4; 3, 3; 5, 2; 7, 1; 13, 1; 19, 1] [2161, 1; 8641, 1]
77805001 [2, 3; 3, 2; 5, 4; 7, 1; 13, 1; 19, 1] [1801, 1; 43201, 1]
?
In the two last solutions, the two $n-1$-numbers share the same prime factors, but the first solution is completely different with a squarefree $n-1$-number.
Can we somehow classify all the solutions of this equation ?
What, if we replace $5$ by another positive integer ?