Let $\lambda = (\lambda_1,\lambda_2,\ldots,\lambda_p)$ be a partition of a positive integer $n$. We call $\lambda_i$ a part of $\lambda$. I am interested in the sum of arbitrary parts of $\lambda$.
Whether a formula or an algorithm computes all possible sum of arbitrary parts of $\lambda$.
For example, $\lambda = (1,2,2,3)$, all possible sum of arbitrary parts is $\{1,2,3,4,5,6,7,8\}$.