Curtz's denominator conjecture: OEIS A189733, revision 12, COMMENTS, May 23, 2011

We prove Paul Curtz's six-residue denominator conjecture in OEIS A189733, revision 12, COMMENTS (May 23, 2011), now in revision 36, FORMULA, with its convention that zero has denominator one. Following Gao's earlier generating-function proof, we derive exact values of the first row of the defining rational array. These values also prove the six-periodic numerator assertion, the denominator recurrence, and Barker's empirical generating function in the same entry. The same formulas settle Barker's recurrence and generating-function conjectures in A267068 and give the numerator identities in A278718. We also prove the full-array reflection and binomial-transform identity. The corresponding second-kind array yields A194767's denominator formulas and reciprocal sum, and corrects an index in its fourth-diagonal numerator identity. For the unsigned-seed array in A189731 and A242926, the same argument gives a Lucas-number formula and refutes the asserted prime-only integer ranks at the composite 705. An integer triangular transformation determines antidiagonal denominator lcms for arbitrary rational seeds.