Minimal first occurrence of each row-count value

Determine whether, for every integer v≥2, the least positive integer N such that N belongs to exactly v rows R_m={⌊kφ^m⌋:k≥1} is the Lucas number L_{4v−5}.

Background

The paper defines R_m={⌊kφ^m⌋:k≥1} and a(N)=#m≥1:N∈R_m, the number of rows of Kimberling’s array containing N. The authors prove that every positive integer v occurs infinitely often as a value of a(N), and that each level set {N:a(N)=v} has positive natural density.

Proposition 4.1 establishes that a(L_m)=(m+1)/4+1 for odd m≥3, implying a(L_{4v−5})=v for every v≥2. The unresolved issue is whether these explicitly identified Lucas numbers are not merely examples with row count v, but are the smallest such examples. Computations through N≤10^8 verify the claim for v≤10.