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First, for a given n, each player partitions the sequence of
the first n positive integers into the two sets {a(k)} and {b(k)}.

A player's score is the number of positive divisors of the sum:

product a(k)'s + product b(k)'s.

So, for example, if n = 6, a player may have the sum of the products:

1*3*4*5 + 2*6 = 72.

Since 72 has 12 positive divisors, the player gets 12 points.
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Allowing for empty sets (so a(1) can be defined), I get that
the maximum scores sequence begins:
2, 2, 2, 4,...