On the problem σod(n) = σod(n+1)

Let ?od(n) = ? d|n,2|?dd. In this paper, we study the solutions of ?od(n) = ?od(n+1), their relations to Pell numbers, and some interesting conjectures. Finally, we obtain that the equation ?od(n) = ?od(n + 1) = ?od(n + 2)?1 (mod 2) has no solution.