...SPOILER TAGS

7, 2, 5, 5, 6, 0, 8, 3, 0, 6, 9, 0, 2, 1, 7, 8, 4, 8, 6, 5, 3, 9, 4, 2, 0, 6, 4, 2, 7, 7, 1, 8, 0, 0, 2, 7, 5, 0, 0, 0, 5, 7, 1, 9, 9, 1, 3, 0, 1, 9, 4, 7, 3, 3, 0, 0, 9, 3, 8, 5, 2, 5, 2, 9, 9, 2, 0, 0, 6, 9, 9, 8, 9, 9, 1, 3, 1, 4, 8, 3, 8, 6, 0, 9, 3, 6, 9, 8, 0, 8, 4, 4, 3, 4, 9, 7, 6, 4, 7, 9

Take the smallest multiple of 3, add the largest, and multiply by 166.5.
Take the smallest multiple of 5, add the largest, and multiply by 99.5.
Subtract the sum of the common multiples.

For Co(n), let the maximal sum subset be denoted as S.
Each prime factor below n must be found in the factorization of exactly one term in S. As such, we can start off the problem with a base case by creating a subset of all the primes below n (and 1) and then raising each of these primes to the highest power possible to maximize the sum.

However, it is possible that a term has the form (p^a)(q^b), where p and q are primes, if (p^a)(q^b) > p^c + q^d. (where a,b,c,d are the highest possible powers such that each term is less than n)

So the problem becomes reduced to figuring out how to find all of the optimizations for the sum. Chances are, there's some pattern that can use the fact that they give you the values of Co(30) and Co(100).

For Co(10):
sorted mainlist [1, 5, 7, 8, 9]
sum of mainlist 30
primes [3, 5, 7]

this is fine

For Co(30):
sorted mainlist [1, 11, 13, 17, 19, 23, 25, 27, 28, 29]
sum of mainlist 193
primes [3, 5, 7, 11, 13, 17, 19, 23, 29]

this is correct too

But for Co(100):
sorted mainlist [1, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 98]
sum of mainlist 1248
primes [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

this is incorrect

sorted mainlist [1, 7, 11, 13, 16, 17, 19, 23, 25, 27, 29]
sum of mainlist 188
primes [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]

This is what I get for Co(30) without that optimization and going for power-raising alone. Obviously I need to toss 28 into this beast, but in doing so I must remove 7 and 16.

EDIT: What I do now, in general is create the "naive prime-raised" solution and then add numbers to the solution and remove all coprime elements, then check to see if that sum is better than what I had before. If so, update solution. If not, kept what I had before and try a new number.

sorted mainlist [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97, 99]
sum of mainlist 1355
primes [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

Could this be extended to cover cases of (p^a)(q^b)(r^c) > p^d + q^e + r^f etc. ? I can't think of an example though. The huge annoyance here is that the list of numbers are exhaustive, everytime you form a set of p,q there could be another set which may use them more efficiently instead (pointing out the obvious though)

I have a conjecture that a term of the form (p^a)(q^b)(r^c) is never found in a maximal sum subset, but I'm not sure if I can prove it. If this is true it would at least make the problem significantly easier. I forgot to mention this earlier.

Since you are one off from the actual answer (changing from odd to even), it implies that one set of (p^a)(q^b) should be degrouped, or grouped (e.g. 5*19=95(odd), or 5^a+19^b=even). Swapping pairs of number products will not change the odd/even parity
The main concern are the primes <10 in this case (because obviously you can't multiply two primes >10 that will result in <100 product)
So the usable primes are 2,3,5,7
Since you used up all 4 and ended up with 1355; If 1356 was not a mistake, that means that you have to degroup one of those primes and leave one as a natural power of itself (which is counterintuitive)
...So either the 13 people who solved didnt bother checking Co(100), or I'm missing something rofl
(or a rare case of a possible product of three primes, which idk if its plausible)

For a possible general approach, to try to maximise sum, we can start by getting rid of all the small primes
So e.g. for Co(100), start with 11,13,17,19. With your current solution we got: 99 (3,11), 95 (5,19), and 92 (2,23), 91 (7,13). Could there be a possible solution using 17 in place of 23?

initial naive list from prime-raising alone: [1, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97] with sum 1263

adding 99 to the list and removing non-coprimes to test [1, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 99]
Updating list because biggestsum 1263 is less than 1270
adding 98 to the list and removing non-coprimes to test [1, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 98, 99]
adding 96 to the list and removing non-coprimes to test [1, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 96, 97]
adding 95 to the list and removing non-coprimes to test [1, 13, 17, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 95, 97, 99]
Updating list because biggestsum 1270 is less than 1321
adding 94 to the list and removing non-coprimes to test [1, 13, 17, 23, 29, 31, 37, 41, 43, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 94, 95, 97, 99]
adding 93 to the list and removing non-coprimes to test [1, 13, 17, 23, 29, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 93, 95, 97]
adding 92 to the list and removing non-coprimes to test [1, 13, 17, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
Updating list because biggestsum 1321 is less than 1326
adding 91 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97, 99]
Updating list because biggestsum 1326 is less than 1355
adding 90 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 90, 91, 97]
adding 88 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 88, 89, 91, 95, 97]
adding 87 to the list and removing non-coprimes to test [1, 17, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 92, 95, 97]
adding 86 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 86, 89, 91, 95, 97, 99]
adding 85 to the list and removing non-coprimes to test [1, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 91, 92, 97, 99]
adding 84 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 84, 89, 95, 97]
adding 82 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 82, 83, 89, 91, 95, 97, 99]
adding 81 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 92, 95, 97]
adding 80 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 80, 83, 89, 91, 97, 99]
adding 78 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 78, 79, 83, 89, 95, 97]
adding 77 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 92, 95, 97]
adding 76 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 76, 79, 83, 89, 91, 97, 99]
adding 75 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 91, 92, 97]
adding 74 to the list and removing non-coprimes to test [1, 17, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 74, 79, 83, 89, 91, 95, 97, 99]
adding 72 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 72, 73, 79, 83, 89, 91, 95, 97]
adding 70 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 70, 71, 73, 79, 83, 89, 97, 99]
adding 69 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 69, 71, 73, 79, 83, 89, 91, 95, 97]
adding 68 to the list and removing non-coprimes to test [1, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 68, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 66 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 66, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 65 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 92, 97, 99]
adding 64 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 63 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 63, 67, 71, 73, 79, 83, 89, 92, 95, 97]
adding 62 to the list and removing non-coprimes to test [1, 17, 29, 37, 41, 43, 47, 53, 59, 61, 62, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 60 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 71, 73, 79, 83, 89, 91, 97]
adding 58 to the list and removing non-coprimes to test [1, 17, 31, 37, 41, 43, 47, 53, 58, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 57 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 57, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 56 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 55 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 54 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 54, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 52 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 52, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 51 to the list and removing non-coprimes to test [1, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 50 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 50, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 49 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
adding 48 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 48, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 46 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 46, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 45 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 45, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 44 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 44, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 42 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 42, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97]
adding 40 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 40, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 39 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 39, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97]
adding 38 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 38, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 36 to the list and removing non-coprimes to test [1, 17, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 35 to the list and removing non-coprimes to test [1, 17, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 97, 99]
adding 34 to the list and removing non-coprimes to test [1, 29, 31, 34, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 33 to the list and removing non-coprimes to test [1, 17, 29, 31, 33, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 32 to the list and removing non-coprimes to test [1, 17, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 30 to the list and removing non-coprimes to test [1, 17, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97]
adding 28 to the list and removing non-coprimes to test [1, 17, 28, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 27 to the list and removing non-coprimes to test [1, 17, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 26 to the list and removing non-coprimes to test [1, 17, 26, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 25 to the list and removing non-coprimes to test [1, 17, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97, 99]
adding 24 to the list and removing non-coprimes to test [1, 17, 24, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 23 to the list and removing non-coprimes to test [1, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 22 to the list and removing non-coprimes to test [1, 17, 22, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 21 to the list and removing non-coprimes to test [1, 17, 21, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97]
adding 20 to the list and removing non-coprimes to test [1, 17, 20, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 19 to the list and removing non-coprimes to test [1, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97, 99]
adding 18 to the list and removing non-coprimes to test [1, 17, 18, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 16 to the list and removing non-coprimes to test [1, 16, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 15 to the list and removing non-coprimes to test [1, 15, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 14 to the list and removing non-coprimes to test [1, 14, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 13 to the list and removing non-coprimes to test [1, 13, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
adding 12 to the list and removing non-coprimes to test [1, 12, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 11 to the list and removing non-coprimes to test [1, 11, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 10 to the list and removing non-coprimes to test [1, 10, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 9 to the list and removing non-coprimes to test [1, 9, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 8 to the list and removing non-coprimes to test [1, 8, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 7 to the list and removing non-coprimes to test [1, 7, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
adding 6 to the list and removing non-coprimes to test [1, 6, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 5 to the list and removing non-coprimes to test [1, 5, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97, 99]
adding 4 to the list and removing non-coprimes to test [1, 4, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 3 to the list and removing non-coprimes to test [1, 3, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 2 to the list and removing non-coprimes to test [1, 2, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]

sorted, final mainlist [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97, 99]
sum of mainlist 1355

Your solution is too linear. You should take into consideration the possible case that immediately adding the largest possible number won't optimize the overall total.

It's possible there is a different combination of the primes that rounds out to a better sum. I recommend this method:

- Separate primes into 3 categories: between 2 and sqrt(n), between sqrt(n) and n/2, and the others
- Add up all of the primes in the last category. These don't change.
- The terms in the second category have the most room for possible optimizations, and therefore should try to be optimized first. They obviously cannot be paired with a prime from its own category, so pair it with a prime from the first category. Try every combination here.
- The last possible case is that 2 primes from the first category might be paired together. Consider this generalization last.

Do it this way and see if you can get to 1356.


NOTE: I'm not sure if the last step is necessary, I'm trying to see if I can prove whether or not that last step is needed.

For 100, we have 25 odd primes (including 1), and one even 2. The sum of 25 odd primes (raised to any power) sum up to odd, adding the power of 2 still makes it odd

However, since we can group primes, we most likely will be grouping the smallest primes. There are 4 primes <sqrt100, 2 3 5 7. By grouping, we have 22 odd numbers and 2. 2 can be combined with an odd number leaving 21 odd numbers and one even number. The sum of that is odd. 1356 is even (?????????)

Possible explanation would be that the solution for 1356 involves -not- grouping one of 3, 5 or 7 and leaving it as its natural power. The most plausible is 3^4=81. Alternatively, there may be a case where there is a coprime which is the product of 3 primes

{1, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 88, 89, 91, 95, 97}
i went off the fact that 81 is the largest number less than 100 of the form a^b

Okay, here's my updated algorithm. It seems to be pretty tight for the most part, but it is still fairly brute force and assumes two conjectures:

- A term in a maximal sum subset can never have more than two unique prime factors.
- When optimizing the sum of a subset by choosing to multiply two powers of primes, the two primes are never both below sqrt(n).

Assuming these conjectures, we use the method:
- Separate primes into 3 categories: between 2 and sqrt(n), between sqrt(n) and n/2, and the others
- Add up all of the primes in the last category. These don't change.
- Each of the terms in the first category can either be left by itself or paired with a term from the second category. Consider and test every case.

The total number of possible subsets to test for is exactly 9507!/(9421)!.

Hopefully we can do a little bit better than that, as this number is in the order of about 10^(well over 200).

2L^2/3 = 2*3^(any non-negative integer)*n^2*(6p+1)^4*(6q+1)^4*(6r+1)^2
n must only be a multiple of 2 and any primes of the form 6k-1
6p+1, 6q+1 and 6r+1 are distinct prime numbers; p, q and r are integers

5 : [4, 3, 5]
6 < 4+3, don't insert
6 : [4, 3, 5]
7 has one prime factor, insert
7 : [4, 3, 5, 7]
8 has one prime factor, insert
8 : [8, 3, 5, 7]
9 has one prime factor, insert
9 : [8, 9, 5, 7]
10 < 8+5, don't insert
10 : [8, 9, 5, 7]
11 has one prime factor, insert
11 : [8, 9, 5, 7, 11]
12 is skipped, prime factors are too small
12 : [8, 9, 5, 7, 11]
13 has one prime factor, insert
13 : [8, 9, 5, 7, 11, 13]
14 < 8+7, don't insert
14 : [8, 9, 5, 7, 11, 13]
15 > 9+5, insert
15 : [8, 15, 15, 7, 11, 13]
16 has one prime factor, insert
16 : [16, 15, 15, 7, 11, 13]
17 has one prime factor, insert
17 : [16, 15, 15, 7, 11, 13, 17]
18 is skipped, prime factors are too small
18 : [16, 15, 15, 7, 11, 13, 17]
19 has one prime factor, insert
19 : [16, 15, 15, 7, 11, 13, 17, 19]
20 gets a bit more complex
20 drags in 16 and 15, but 15 is also a multiple of 3
it quickly becomes apparent that the only thing one could do with the 3 is use 9
20+9 < 16+15, don't insert
20 : [16, 15, 15, 7, 11, 13, 17, 19]
21+5 > 15+7, insert
[16, 21, 5, 21, 11, 13, 17, 19]
20 < 16+5, don't attempt
21 : [16, 21, 5, 21, 11, 13, 17, 19]
22 < 16+11, don't insert
22 : [16, 21, 5, 21, 11, 13, 17, 19]
23 has one prime factor, insert
23 : [16, 21, 5, 21, 11, 13, 17, 19, 23]
24 is skipped, prime factors are too small
24 : [16, 21, 5, 21, 11, 13, 17, 19, 23]
25 has one prime factor, insert
25 : [16, 21, 25, 21, 11, 13, 17, 19, 23]
26 < 16+13, don't insert
26 : [16, 21, 25, 21, 11, 13, 17, 19, 23]
27 has one prime factor, insert, 7 is left over
[16, 27, 25, 7, 11, 13, 17, 19, 23]
14 < 16+7, don't attempt
27 : [16, 27, 25, 7, 11, 13, 17, 19, 23]
28 > 16+7, insert
28 : [28, 27, 25, 28, 11, 13, 17, 19, 23]
29 has one prime factor, insert
29 : [28, 27, 25, 28, 11, 13, 17, 19, 23, 29]
30 is skipped, too many prime factors
30 : [28, 27, 25, 28, 11, 13, 17, 19, 23, 29]

#1. I think because he's listing the numbers in order of smallest prime factors, and some numbers are made up of two prime factors so he repeated them
#2. Probably "set 1" (i.e. primes below sqrt n for Co(n), using the product of two numbers from set 1 would be a waste

I had this thought, but I don't know if its useful. See if anyone finds a use of it.
As Stargroup listed 3 sets of numbers, for Co(n^2), where set 1 is for primes<n and set 2 between n and (n^2)/2. I propose listing numbers of set 1 in the form of n-a, and set 2 in the form n+b

For products not involving powers higher than 1 for each term, we have to fulfill this condition (n-a)(n+b)≤n^2 [e.g. for Co(100), (10-5)(10+9)<10^2]

Expand to n^2-na+nb-ab≤n^2

hence n(b-a)≤ab (but we don't know if a or b is bigger or smaller)
or b≤na/(n-a)

ok now what

i have no idea why you're confused
what's going on is i give the program a number and its one or two prime factors
if there's 1, it automatically goes into the list (and the consequences of said action are played out)
if there's 2, i retrieve the numbers in the list that share the two factors
so for 20, i grab a_2 and a_5, while for 21, i grab a_3 and a_7
the idea is we're seeing if a set with n in it produces a larger sum than just copying over the previous set
of course, i can't forget that one or both of the numbers i retrieved might be duplicates
in both cases, i find that a_3 = a_5

i now need to find whether i can shuffle around the three primes ((2, 3, 5) for 20, (3, 5, 7) for 21) and produce a sum larger than that of the a_i i retrieved

for 20, i already know i'm using 2 and 5 as part of 20, so i need to figure out what to do with 3
i could simply choose to multiply it by itself, giving 3*3 = 9
so now i check to see if (a_2, a_3, a_5) = (20, 9, 20) is better than (a_2, a_3, a_5) = (16, 15, 15); in the quote i find it is not
i can't multiply 3 by 7 because 21 is larger than 20, so i am done

as for 21, i already know i'm using 3 and 7 as part of 21, so i need to figure out what to do with 5
i could simply leave it as 5, so i'm testing if (a_3, a_5, a_7) = (21, 5, 21) is better than (15, 15, 7); in the quote i find that it is
however, there's also the possibility that i could combine a_2 and a_5, making both 20; in the quote i find it doesn't work, and i can't mess with a_3 now since i changed it, so i am done

17 is brought back as part of Co(95)
95+17 > 85+19
it's the exact same phenomenon that occurs with 21 and 5 replacing 15 and 7

Stargroup originally came up with the idea that having a composite being a product of 3 primes isn't efficient and will never be part of the maximum set. I had a thought and will back this up by saying that, in order for a composite to be in the set it has to have at least two terms from group 1 (otherwise it will exceed n). And it is intuitively known to be not efficient to group together two sets of numbers from set 1 (as you remove the chance of it being paired up with a set 2 number, which is more efficient to produce larger composites. So I'll assume that the solution set involves composites with at most two distinct primes (raised to whatever power). So in this case, 30 which has 3 prime factors will not be part of the solution set of Co

I suck at this problem pretty hard, admittedly. It's always a split between making some assumption about the composite set (which yields the wrong answer) or trying all possible combinations (too large to run in this lifetime) -- the same problem that's been knocking at me from the beginning of this damn problem -_-