I believe an upper bound for f(n) is given by https://oeis.org/A092249 which can be interpreted as the number of segments resulting from slicing the interval [0, 1] into 2, 3, ... n uniform segments. Clearly this satisfies all of the required sums. The first several values are: 1,2,4,6,10,12,18,22,28,32,42,46,58,64,72,80,96,... Note that this is also https://oeis.org/A002088 but without the initial zero entry. The question is, is it ever possible to do better than this? Tom James Propp writes:
Perhaps my question has been considered in the past as a question about cutting an interval into pieces, since the circularity of the pie/pizza/whatever is irrelevant. (The very first radial cut effectively turns a problem about cutting a disk into wedges into a problem about cutting an interval into subintervals.)
Or maybe we should get rid of geometry entirely, and just ask: What is the smallest collection of fractions (with repetitions allowed), summing to 1, such that by combining fractions in the collection we can write 1 as 1/2 + 1/2, or as 1/3 + 1/3 + 1/3, or ..., or as 1/n + 1/n + ... + 1/n?
Let f(n) be the smallest possible number of such fractions. Clearly f(1) = 1 and f(2) = 2, and it's not hard to show that f(3) = 4. I haven't figured out f(4) (it's either 5 or 6). Has anyone seen this sequence before?
Jim