$24$ cannot be written in such form Let there is number $n>24$ that cannot be written in such form and $m^2 \leq n <(m+1)^2$ If $m-1\not |n$ then $a \equiv n \mod{(m-1)},b=m-1,c=\frac{n-a}{b}\geq \frac{m^2-(m-2)}{m-1}>m$ so $n=a+bc$ and $a<b<c$ So $m-1|n$ and as $m^2-1<n<(m+1)^2 \leq (m-1)(m+4)$ for $m \geq 5$ then $n=(m-1)(m+2)$ or $(m-1)(m+3)$ But for $n=(m-1)(m+3)=m^2+2m-3=(m-3)+m(m+1)$ and if $m>6$ then $n=(m-1)(m+2)=m^2+m-2=4+(m-2)(m+3)$ So enough to check numbers $n=4*7=1+3*9$ and $n=5*8=1+3*13$