Reutenauer and Kassel introduced a family P_n(q) of polynomials defined in terms of divisors of n on overlapped intervals. The evaluation of P_n(q) at roots of unity of order 2, 3, 4, 6 form well-known integer sequences related to the number of integer solutions of the equations x² + y² = n, x² + 2y² = n, and x² + xy + y² = n. Also, P_n(1) is the sum of divisors of n. In this paper we define a new family L_n(q) of polynomials defined in terms of divisors of n on overlapped intervals, slightly modifying the definition of P_n(q). The values of L_n(q) at q = 1 and q = -1 are related to the sum of divisors of n and to the number of integer solutions of the equations x² + xy + y² = n and x² + 3 y² = n.