Values of v(k), the smallest denominator missing from a k-term unit fraction decomposition of 1

Abstract

For a positive integer k let Dₖ be the set of integers that occur as a denominator in some identity 1=1/n₁+…+1/nₖ with 1 ≤ n₁<…<nₖ, and let v(k) be the smallest integer larger than 1 not in Dₖ. Erdős and Graham introduced v(k) in 1980 and asked how fast it grows; van Doorn and Tang have recently proved the first lower bound, v(k) ≥ e^(ck²), and their paper prints no value of v(k) beyond D₂=emptyset and D₃={2,3,6}. We determine v(k) for every k ≤ 8, namely 2,2,4,11,17,103,733,27539. We have found no source that states v(6)=103, though a 2020 comment on the OEIS sequence A330808 already implies its non-membership half; v(7)=733 and v(8)=27539 are the seventh and eighth terms of the OEIS sequence A097048, which records data on the neighbouring problem of Erdős and Graham on the length N(a,b) of Egyptian fraction expansions. All eight values are proved here rather than quoted. We also compute |Dₖ| for k ≤ 6, namely 1,0,3,15,77,1152, show that Dₖ is finite for every k, and prove the upper bound v(k) ≤ |Dₖ|+2 in general. Everything rests on a bounded depth-first search that is proved exactly correct — its output is the set of decompositions, not merely a subset of it — so that an empty search is a non-existence theorem. The proofs are machine-checked in Lean 4; Section [sec:verif] records the scope of the formalisation and which statements are checked by the proof kernel alone.

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