I have to find algorithm which find prime number less than $n$ which is sum of the largest amount of unique primes, for example for $n=81$, the answer is $79 = 3 + 5 + 7 + 11 + 13 + 17 + 23$. I have read about partitions and Goldbach's conjecture but it seems unhelpful.
After the first split of $n$ into two or three unique primes, the problem becomes splitting larger primes into a list of smaller primes which is the problem I discuss below.
The minimum number of unique primes a larger prime can be split into is $2$ smaller primes, where $2$ is one of those unique primes e.g. $31=29+2$. If $2$ is not used the minimum number of primes a larger prime can be split into is $3$ primes e.g.
We can now look at how to generate new primes by making the split into $3$ other primes and so on. The split into $2$ primes using $2$ can only be used once.
e.g. the smaller primes above split as follows
$$29=(19+7+3)=(17+7+5)=(13+11+5)$$ $$23=(13+7+3)=(11+7+5)$$ $$19=(17+2)=(11+5+3)$$ $$13=(11+2)$$ $$7=(5+2)$$
In the example of $31$ I first split this into 2 primes because that gives the largest available prime, that is $31=29+2$
I then use the split $29=(19+7+3)$ giving
The question is:
Is this "greedy prime" algorithm (taking the largest primes you can get from any recursive split into 2 or mainly 3 primes) always going to get you to a sum with the largest number of unique primes?
[Which includes the constraint to always decompose the largest prime in the new list first (the new list being generated immediately after a previous split).]