set with maximum sum consisting of mutually co-prime numbers

Question

Definitions. Let $n$ be a natural number and $S$ be a subset of distinct natural numbers all less than $n$, and mutually co-prime. Then find the maximum sum the set $S$ can have.

Example. Let $n=10$, then the maximum sum $S$ can have is $30$ corresponding to $S=\{1,5,7,8,9\}$.

What I tried. I tried to implement the following greedy algorithm.

1. First I divide primes $\le n$ into $3$ lists: $list_1$ containing primes $\le \lfloor \sqrt{n} \rfloor$, $\; list_2$ containing primes in range $\big( \lfloor \sqrt(n) \rfloor,\lfloor \frac{n}{2} \rfloor \big )$ and finally $list_3$ containing prime in the range $\big[\lfloor \frac{n}{2} \rfloor+1,n \big ]$.
2. Next I define a function $largest(a,b,n)$, which returns the largest number of the form $a*b^{r}$ less than equal to $n$.
3. If there exist two numbers $l_1 \in list_1$ and $l_2 \in list_2$ such that $diff = largest(l_2,l_1,n)-( largest(1,l_1,n) + l_2 ) > 0$, I chose the pair $(l_1,l_2)$ such that $diff$ is maximised. I delete $l_1$ from $list_1$, $l_2$ from $list_2$ and append $largest(l_2,l_1,n)$ in $list_2$ and repeat step $3$. If no such pair $(l_1,l_2)$ exists such that $diff>0$, I go to step $4$.
4. I don't change anything in $list_3$.
5. In the end $ \{1\} \cup \{ l_1^r \; | \;l_1 \; \in list_1 \text{ and } largest(1,l_1,n)=r \} \cup list_2 \cup list_3$ is my desired set $S$ and I report the sum of it's elements.

Running example of my algorithm. If $n=30$,

#list_1 = {2,3,5}
#list_2 = {7,11,13}
#list_3 = {17,19,23,29}

#The first time step 3 is evaluated I get l_1 = 2 and l_2 = 7 ..
#.. as  diff = ( 28 - ( 16 + 7 ) ) = 5 > 0 and diff is maximised
# so I delete 2 from list_1 and 7 from list_2 and append 28 in list_2

#When step 3 is evaluated again no such pair (l_1,l_2) is found

#So our desired list becomes S={1,27,25,11,13,28,17,19,23,29} with sum 193.    

But my algorithm only seems to give correct for few simple cases only. Plus I am not able to prove/disprove or modify any of my assumptions .Now I am hopelessly stuck at the problem and I am not making any progress. But I still believe that some greedy algorithm is at work here.

PS: The question is from project Euler max-sum co-prime set. I would really appreciate just some hint or new direction to think.

Answer 1

Project Euler asks you to solve the problems yourself, without help. So dont read on if you want to submit a solution for Project Euler; that would be cheating.

Since the numbers are mutually co-prime, each prime number p is a factor of at most one element of S. On the other hand, if p is small enough then a number could have a factor $p^2$, $p^3$, $p^4$ and so on. We also know that every prime number p ≤ N is used (if p is not used as a factor of any number in the set then we just add p). And don't forget to include the number 1 in the set S :-)

A number x ≤ N cannot have two prime factors greater than $N^{1/2}$. Therefore the set S contains among other numbers one number $x_p$ ≤ N which is a multiple of a prime $p > N^{1/2}$. That's a good start for finding the optimal set S. We then take the primes $q ≤ N^{1/2}$, and either multiply one of the numbers $x_p$ by a power of q, or we add a new number $x_q$ which is some power of q.

In the Project Euler project with N = 200,000, there are fewer than 90 primes $q ≤ N^{1/2}$. We can take each of these primes q in turn, and either multiply one of the existing $x_p$ by a power of q, or add a new number $x_q$. We would try to do this achieving the highest possible increase in the sum. If no two q, q' achieve the highest increase in the same way, we can pick the optimal way for each.

In the example N = 30, we would start with $x_p$ = 7, 11, 13, 17, 19, 23, 29. We can use q = 2 to change 7 -> 28. q = 3 -> 27, q = 5 -> 25. The gains are 21, 27, 25. Each gain is > N/2. So an optimal solution cannot include a number created by two or more of the q's - that number would be ≤ N, but we would give up two gains > N/2, so this wouldn't be optimal. So in the simple example, we multiply 7 by $2^2$ and add $3^3$ and $5^2$.

With N = 100, we would have $x_p$ = 11, 13, 17, 19, 23, ... and q = 2, 3, 5, 7. q = 2: 11 -> 88 gains 77. q = 3: 11 -> 99 gains 88. q = 5: 19 -> 95 gains 76. q = 7: 13 -> 91 gains 78. Unfortunately we have a conflict; q = 2 and q = 3 would both use $x_{11}$. If we use 11->88 (gain 77), then q = 3: 31->93 with a gain of 62; total gain 139. If we use 11->99 (gain 88) then q = 2: 23->92 (gain 69) has a total gain of 157. So optimal for N = 100 is

S = { 1, 99, 91, 17, 95, 92, 29, 31, 37, 41, ,,, }

So here is the algorithm: Start with $x_p$ = primes > $N^{1/2}$. Find the primes $q ≤ N^{1/2}$. For each q, find the maximum gain that can be achieved using q: Either $q^k$ by adding the number $q^k$, or $p * (q^k - 1)$ by multiplying $x_p$ by $q^k$. If each optimal gain is achieved in a different way, and each gain is ≥ N/2, then we found the optimal solution.

Otherwise, we choose one set of q's that use the same $x_p$. For each of these q's: We assume that this q uses $x_p$, find the optimal solution under the restriction that no other q uses this $x_p$, and choose which q using $x_p$ gives the best total.

If there is a case where the two smallest gains add up to less than N, ask someone for a more complex solution :-)

Answer 2

I did not follow gnasher729 completely on how he resolved the conflicts he mentions in his answer ( will look into it now ). So as suggested by D.W. I spent more on it and finally solved it. I followed the same algorithm described in my question except for the fact, that in place of doing greedy, one has to use maximum weighted bipartite matching ( a suggestion by Evil ), while trying to pair up a prime less than $\sqrt(n)$ with a prime greater than $\sqrt(n)$. But as I did not know how to implement maximum weight bipartite matching, I looked closely and the bi-partite graph formed in this case was too easy and one could resolve the conflicts and thus solve for maximum weight manually ( by pen and paper calculator :), the bipartite graph formed had interesting patterns too ). Finally the solution rests on a conjecture I am still unable to proof ( which I assumed in my post ) : to increase the sum of set $S$ you can pair up a prime less than $\sqrt{n}$ with exactly one prime greater equal to $\sqrt{n}$.