Any integer greater than 31 can be so represented. If an integer $N$ is composite, and $p$ is the smallest prime which divides $N$, we can simply choose any four integers that add up to $N/p$ and then multiply through by $p$. For prime $N >= 37$, if $N \equiv 1\pmod{6}$, then we can use $\{6a, 6, 10, 15\}$ for the suitable value of $a$. And starting with $N = 41$, if $N \equiv 5\pmod{6}$, we have $\{6b, 10, 10, 15\}$ for suitable $b$.