Let $C_k$ be the largest positive integer that cannot be written as the sum of $k$ distinct primes for $k\geq 4$. Using that all odd integers greater than 17 can be written as the sum of 3 distinct primes (which I believe is proven in Harald Helfgott's proof of the ternary Goldbach Conjecture) one can use an algorithm to find $C_4$, then $C_5$, then $C_6$ etc and this can be used to inductively show that $C_k$ exists and is finite for all $k\geq 4$. The Oeis sequence A124884 lists $C_k$ up to $k=50$.
As $2$ is the only even prime, if $k$ is odd then the only way to an odd number as the sum of $k$ distinct primes is if all $k$ of those primes are odd. Similarly, if $k$ is even then the only way to write an even number as the sum of $k$ distinct primes is if all $k$ of those primes are odd. So letting $p_i$ be an increasing list of the odd primes, $C_k \geq \sum_{i=1}^k p_i$ and it can be straightforwardly argued that in fact $C_k \geq \sum_{i=1}^k p_i+2$ for $k\geq 4$.
It seems clear to expect that $C_{k-1}<C_k$ however I have not been able to prove this. I have tried setting up some kind of induction where I assume this holds up to $k=n$, then consider a number $t$ that can be written as the sum of $n+1$ distinct primes and try to prove that $t$ can be written as the sum of $n$ distinct primes, however I have not been able to make this work. I was wondering if anyone had any ideas for proving this. Additionally, the terms in the Oeis sequence all satisfy $C_k < \sum_{i=1}^{k+1}p_i$ which would imply that $C_{k-1}<C_k$.
