$b=p*a$ $c = pq*a$ $a+ka+d*k*a= n$ $a(1+p+pq) = n$ $p,q \geq 2$ $pq+p+1 \geq 7$ $n \ne 1,2,3,4,5,6$ for $p =2$ $a(3+2p) = n$ letting $a = 1$ all $ n \geq 7 $ odd achievable now the even cases notice if we make $a =2^k$ then all $2^k* (pq+p+1) = 2^k*s $ where $s\geq 7$ is achievable now we only need to check case $n=2^k,2^k*3,2^k*5$ notice $pq+p+1 = 16$ have sols (3,4) thus only $k \leq 3$ that need a further check which is $10,12,20,8,24,40$ $a+b+c = 10$ achievable letting a =1 and $(p,q) = 3,2$ thus implies 20,40 also achievable now check 8,12,24 and then prove that it isnt achievable so all $n \in \mathbb{N} \backslash [1,2,3,4,5,6,8,12,24]$ is the possible answer