Recall that an integer is $t-$free iff it is not divisible by $p^t$ for some
prime $p.$ We give a method to check Robin inequality $\sigma(n) < e^\gamma
n\log\log n,$ for $t-$free integers $n$ and apply it for $t=6,7.$ We introduce
$\Psi_t,$ a generalization of Dedekind $\Psi$ function defined for any integer
$t\ge 2$ by $$\Psi_t(n):=n\prod_{p \vert n}(1+1/p+\cdots+1/p^{t-1}).$$ If $n$
is $t-$free then the sum of divisor function $\sigma(n)$ is $ \le \Psi_t(n).$
We characterize the champions for $x \mapsto \Psi_t(x)/x,$ as primorial
numbers.