Let $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ be the prime decomposition of the integer $n$. Define $n' = n \sum_{i=1}^r \frac{\alpha_i}{p_i}$ and $\Omega(n) = \sum_{i=1}^r \alpha_i$, $\omega(n) = r$. Let $a,b$ be relatively prime ( $(a,b) = 1$ ) and $c = a+b$. Suppose that $\Omega(c) = min(\Omega(a),\Omega(b),\Omega(c))$. Is it true, that $\Omega((a,a'))+\Omega((b,b'))+\Omega((c,c')) \le \Omega(ab)-1$? From this it would follow that $min(\Omega(a),\Omega(b),\Omega(c)) \le \omega(abc) - 1$

Edit: From the answer given by Kevin Buzzard, one can see, that the first inequality is wrong. It is unclear to me however, if the second inequality is also wrong.

Let $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ be the prime decomposition of the integer $n$. Define $$n' = n \sum_{i=1}^r \frac{\alpha_i}{p_i}\quad\text{and}\quad\Omega(n) = \sum_{i=1}^r \alpha_i\quad\text{and}\quad\omega(n) = r.$$ Let $a,b$ be relatively prime, i.e., $\gcd(a,b)=1$, and let $c = a+b$. Suppose that $$\Omega(c) = \min\bigl\{\Omega(a),\Omega(b),\Omega(c)\bigr\}.$$ Is it true that $$\Omega(\gcd(a,a'))+\Omega(\gcd(b,b'))+\Omega(\gcd(c,c')) \le \Omega(ab)-1?$$ From this it would follow that $$\min\bigl\{\Omega(a),\Omega(b),\Omega(c)\bigr\} \le \omega(abc) - 1.$$

Edit: From the answer given by Kevin Buzzard, one can see, that the first inequality is wrong. It is unclear to me however, if the second inequality is also wrong.

Let $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ be the prime decomposition of the integer $n$. Define $n' = n \sum_{i=1}^r \frac{\alpha_i}{p_i}$ and $\Omega(n) = \sum_{i=1}^r \alpha_i$, $\omega(n) = r$. Let $a,b$ be relatively prime ( $(a,b) = 1$ ) and $c = a+b$. Suppose that $\Omega(c) = min(\Omega(a),\Omega(b),\Omega(c))$. Is it true, that $\Omega((a,a'))+\Omega((b,b'))+\Omega((c,c')) \le \Omega(ab)-1$? From this it would follow that $min(\Omega(a),\Omega(b),\Omega(c)) \le \omega(abc) - 1$

Let $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ be the prime decomposition of the integer $n$. Define $n' = n \sum_{i=1}^r \frac{\alpha_i}{p_i}$ and $\Omega(n) = \sum_{i=1}^r \alpha_i$, $\omega(n) = r$. Let $a,b$ be relatively prime ( $(a,b) = 1$ ) and $c = a+b$. Suppose that $\Omega(c) = min(\Omega(a),\Omega(b),\Omega(c))$. Is it true, that $\Omega((a,a'))+\Omega((b,b'))+\Omega((c,c')) \le \Omega(ab)-1$? From this it would follow that $min(\Omega(a),\Omega(b),\Omega(c)) \le \omega(abc) - 1$

Edit: From the answer given by Kevin Buzzard, one can see, that the first inequality is wrong. It is unclear to me however, if the second inequality is also wrong.