Mean $=13$
Variance $=\frac{9+49+144+ a ^{2}+(43- a )^{2}}{5}-13^{2} \in N$
$\Rightarrow \frac{2 a^{2}-a+1}{5} \in N$
$\Rightarrow 2 a ^{2}- a +1-5 n =0$ must have solution as natural numbers
its $D=40 n-7$ always has $3$ at unit place
$\Rightarrow D$ can't be perfect square
So, a can't be integer.