Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer

In this paper, we give a refined form of a recent factorization of Lambert series. This result allows us to prove new connections between partitions and divisors of positive integers, such as a new formula for the number of divisors of a positive integer as a convolution. Three recurrence relations for computing the number of partitions of a positive integer into distinct parts are rediscovered in this context.