This question is related to numbers found in the OEIS sequence A191837.
In this sequence, $a(2) = 48 = 5 + 7 + 17 + 19$, where the summands of 48 are all prime numbers that are less than or equal to $48/2=24$. The prime factors of 48 are $2^4$ and $3$, and neither $2$ nor $3$ are summands in $a(2)$. Similarly, $a(3) = 108 = 5 + 7 + 11 + 19 + 29 + 37$ and the prime factors of $108$ ($2^2$ and $3^3$) are not summands in $a(3)$. This appears to hold for all fourteen (verified and unverified) integers in the sequence.
Is there some way to prove or disprove the conjecture that for every integer in sequence A191837, the prime factors of $a(n)$ will never be an element of the summands of $a(n)$?