I can agree that with the first partitions 1/2, 1/3, 1/7 that four additional partitions are needed to solve the 732/733 problem. The algebraic identities needed to solve this problem include onlu 366/733 = 1/2 - 1/1466 245/733 = 1/3 + 2/2199
121/733 to be paritioned by two terms,
as well as one other term smaller than 1/733.
That is, I will now go onto the 1/2, 1/4, 1/8 first terms and get back with you all, especially David that listed an 'optimal' solution for 1/2, 1/4. If these fail, I will search a little longer using highly composite first partitions that allow the Hultsch-Bruins:
n/p - 1/A = (nA -p)/Ap
n/p = 1/A + (nA -p)/Ap
method to work, as Fibonacci himself understood in 1202 AD.
Yes, it takes a little time to directly solve this problem even using Fibonacci's indeterminate equation method - that I do not think that I have seen listed among Eppstein's < 40 methods.