Let σ k ( n ) denote the sum of the k-th powers of the positive divisors of n. Erdős and Kac conjectured that the sum α k = ∑ n = 1 ∞ σ k ( n ) n ! is irrational for k ⩾ 1 . This is known to be true for k = 1 , 2 and 3. Fix r ⩾ 1 . In this article we give a precise criterion for 1 , α 1 , … , α r to be Q -linearly independent, assuming a standard conjecture of Schinzel on the prime values taken by a family of polynomials. We have verified our criterion for r = 50 .