Denoting by p n and c n the n th prime number and the n th composite number, respectively, we prove that both the sequence ( x n ) n ≥ 1 , defined by x n = ∑ k = 1 n   ( c k + 1 − c k )   /   k − p n   /   n , and the series ∑ n = 1 ∞   ( p c n − c p n )   /   n p n are convergent.