Beyond Perfect Numbers: The Sum of Divisors Divisibility Problem for σ(n) | n + a

The sum-of-divisors function σ(n) has been studied since antiquity, most often in connection with perfect and abundant numbers, yet its behavior under additive divisibility constraints has not been systematically classified. The paper considers the problem of determining, for a fixed integer a, the positive integers n for which σ(n) | n + a. It is shown that for every fixed integer a ≥ 2, only finitely many positive integers n satisfy this relation. The proof reduces the divisibility condition to a size dichotomy: either n < a, yielding only finitely many possibilities, or σ(n) = n + a, which is equivalent to a fixed-value equation for the sum of proper divisors. It is then shown that this equation admits only finitely many solutions for each fixed a. Special cases are described explicitly. When a = 1, the relation σ(n) | n + 1 holds only for n = 1 and for prime n. When a = 0, the condition reduces to σ(n) = 2n, recovering the classical perfect numbers. For a < 0, the inequality σ(n) > n for all n > 1 excludes all but trivial cases. These results complete the classification of shifted divisibility for σ(n) and close the sequence initiated by analogous investigations of φ(n) and λ(n), identifying σ as the terminal case in which multiplicative divisibility collapses to finiteness.