[math-fun] Repunit Divisor Query

The "lower central (median) divisor of n" (OEIS: A060775) is the largest divisor of n that is less than the square root of n. Empirically, for small k, the lower central divisor of repunit(6k), i.e. (10^(6*k)-1)/9, is (10^k - 1)(10^(2*k) - 10^k + 1)/3. The first few examples are: 273, 326733, 332667333, 333266673333, 333326666733333, etc. Is this ALWAYS so? Alternatively, is there a k for which the lower central divisor of (10^(6*k)-1)/9 is GREATER than (10^k - 1)(10^(2*k) - 10^k + 1)/3?