On a problem on partitions into distinct primes

For a positive integer $n$, we write $L(n)$ in place of the maximum number of distinct primes whose sum is $n$, assuming there is a way of writing $n$ as a sum of distinct primes, with $L(n) = 0$ otherwise. Again if $n$ admits a partition into distinct primes, then we let $K(n)$ denote the number of