On integers of the form [formula omitted].

Let r 1 , ... , r t be positive integers and let R 2 (r 1 , ... , r t) be the set of positive odd integers that can be represented as p + 2 k 1 r 1 + ⋯ + 2 k t r t , where p is a prime and k 1 , ... , k t are positive integers. It is easy to see that if r 1 − 1 + ⋯ + r t − 1 < 1 , then the set R 2 (r 1 , ... , r t) has asymptotic density zero. In this paper, we prove that if r 1 − 1 + ⋯ + r t − 1 ≥ 1 , then the set R 2 (r 1 , ... , r t) has a positive lower asymptotic density. Several conjectures and open problems are posed for further research. [ABSTRACT FROM AUTHOR]