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For a positive integer n, let σ(n) denote the sum of the positive divisors of n. Let d be a proper divisor of n. We call n a near-perfect number if σ(n) = 2n + d, and a deficient-perfect number if σ(n) = 2n - d. We show that there is no odd near-perfect number with three distinct prime divisors and determine all deficient-perfect numbers with at most two distinct prime factors.

Let G be a finite cyclic group. Every sequence S over G can be written in the form $S=\left(n₁g\right)·...·\left(n_{l}g\right)$ where g ∈ G and $n₁,...,n_{l}i\in [1,ord\left(g\right)]$, and the index ind(S) is defined to be the minimum of $(n₁+\cdots +n_l)/ord\left(g\right)$ over all possible g ∈ G such that ⟨g⟩ = G. A conjecture says that every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd(|G|,6) = 1 has index 1. This conjecture was confirmed recently for the case when |G| is a product of at most two prime powers. However, the general case is still open. In this paper, we make some...