Unboundedness and surjectivity of T(n)

Establish that the minimal length T(n) of a corresponding square-product sequence for Ron Graham’s sequence g(n) is unbounded and that T(n) attains every positive integer value except 2; equivalently, for every integer t ≥ 1 with t ≠ 2, identify n ≥ 0 such that T(n) = t, where T(n) denotes the least length t of any increasing sequence a1 < a2 < ⋯ < at beginning at n and ending at g(n) whose product a1a2⋯at is a perfect square.

Background

The paper defines Ron Graham’s sequence g(n) as the least k such that there exists an increasing sequence starting at n and ending at k with square product. It then defines T(n) as the minimal length of such a corresponding sequence. Empirically, the authors computed T(n) < 15 for all n ≤ 10000 and provided explicit examples realizing many small lengths (except 2), motivating the conjecture that T(n) is unbounded and surjective onto the positive integers excluding 2.

This problem asks for a global characterization of the range and growth of T(n), extending computational evidence to a full proof that arbitrarily large minimal lengths occur and that every positive length except 2 is realized.