Consider a set $ A ={1,2,3,3,4,4,.....,n,n,n,...,n}$ .
Such that every k $ \in A$ occurs exactly $ \phi (k)$ times, where ,$\phi :N \to N$ and it denote number of coprimes before it.
[it is euler's totient funtion]
Consider $N_0 =\phi (1)+......+\phi (n)$. Now consider $a_i$ be permutation of the elements of A $ \forall i={1,2,3,.....,N_0}$
Prove that for all n we must have a permutation so that $ \sum_{i=1}^{N_0} \frac{1}{a_ia_{i+1}}$ =1 consider$N_0 +1 \to 1$
What I thought:
We will give proof by induction. Base cases: $$n=1 \implies \{1,1\}$$$$n=2 \implies \{1,2,1\}$$$$n=3 \implies \{1,3,2,3,1\}$$
First, if you have:- $$\frac{1}{ab}=\frac{1}{ax}+\frac{1}{xb} \implies \frac{1}{ab}=\frac{a+b}{xab} \implies x=a+b$$