On the sum of parts with multiplicity at least 2 in all the partitions of n

In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of n providing new combinatorial interpretations for this sum. A connection with subsets of {1, 2,...,n} is given in this context. We provide two different proofs of our results: analytic and combinatorial. In addition, considering the multiplicity of parts in a partition, we provide a combinatorial proof of the truncated pentagonal number theorem of Andrews and Merca.