show that $n$ is prime if an only if $$\phi{(n)}+\sigma{(n)}=n\tau{(n)}$$ for $n>1$
i first suppose that $n$ is not prime , then
$n=\sum\limits_{d|n}{\varphi \left( d \right)}=\sum\limits_{d|n;d<n}{\varphi \left( d \right)}+\varphi \left( n \right)\ge \varphi \left( n \right)+\sum\limits_{d|n;d<n}{1}\ge \varphi \left( n \right)+\tau \left( n \right)+1$
Thus $ \varphi \left( n \right)+\tau \left( n \right)\le n-1$ contradiction
But $\varphi \left( n \right)+\sigma \left( n \right)-\sigma \left( n \right)+\tau \left( n \right)=\tau \left( n \right)\left( n+1 \right)-\sigma \left( n \right)$
is this correct approach to this proof ??
