Sets with even partition functions and 2-adic integers

For P ∈ F2[z] with P (0) = 1 and deg(P ) ≥ 1, let A = A(P ) be the unique subset of N such that ∑ n≥0 p(A, n)zn ≡ P (z) (mod 2), where p(A, n) is the number of partitions of n with parts in A. Let p be an odd prime number, and let P be irreducible of order p ; i.e., p is the smallest positive integer such that P divides 1+ zp in F2[z]. N. Baccar proved that the elements of A(P ) of the form 2km, where k ≥ 0 and m is odd, are given by the 2-adic expansion of a zero of some polynomial Rm with integer coefficients. Let sp be the order of 2 modulo p, i.e., the smallest positive integer such that 2 sp ≡ 1 (mod p). Improving on the method with which Rm was obtained explicitly only when Research supported DGRST of Tunisia, UR 99/15-18, Faculté des Sciences de Tunis.