n = 20
Non-coprimes of current n in previous solution set = [15, 16]
Prime factors of n [2, 5]
Prime factors of non-coprimes of n: [2, 3, 5]
Unique prime factors of non-coprimes not found in n = [3]
does 29 (20+9) beat noncoprime sum 31?
Nope. Do nothing, carry on, people.
n = 20, solution = [1, 7, 11, 13, 15, 16, 17, 19], sum = 99

n = 21
Non-coprimes of current n in previous solution set = [7, 15]
Prime factors of n [3, 7]
Prime factors of non-coprimes of n: [3, 5, 7]
Unique prime factors of non-coprimes not found in n = [5]
does 26 (21+5) beat noncoprime sum 22?
Yes, so add both 21 and 5 to the solution set and remove any noncoprimes.
n = 21, solution = [1, 5, 11, 13, 16, 17, 19, 21], sum = 103