Characterization of inputs achieving g(n) = 2n

Establish that Ron Graham’s sequence g(n) equals 2n if and only if n is prime or n = 6, where g(n) is defined as the least k such that there exists an increasing sequence starting at n and ending at k whose product is a perfect square.

Background

Earlier in the paper, the authors prove the general upper bound g(n) ≤ 2n and show it is tight on all primes greater than 3. They then conjecture that, apart from the exceptional case n = 6, only primes achieve equality, thereby seeking a complete characterization of inputs for which the upper bound is attained.

Resolving this would delineate the precise conditions under which the minimal terminal value equals twice the starting integer, sharpening the understanding of g(n)’s extremal behavior.