A positive integer whose sum of proper positive divisors exceeds the integer itself.

Principle

Identified by the proper-divisor sum s(n) satisfying s(n) > n; organizes integers by an excess of divisorial mass relative to the number.

Demonstration

Example: 12 has proper divisors 1, 2, 3, 4, 6 which sum to 16, so 12 is abundant. Many even numbers are abundant; odd abundant numbers exist but are sparser.

Misapplication

Including the number itself when computing the sum of proper divisors (which would misidentify perfect numbers as abundant) or confusing abundant numbers with weird numbers (which are abundant but not semiperfect).

Consequence

Classification into abundant numbers informs divisor-sum statistics, density results and the study of semiperfect or weird numbers; abundance affects partitioning and combinatorial properties of integers.

Reversal

The opposite concept is deficient numbers (s(n) < n), while perfect numbers lie at the exact boundary (s(n) = n).

Boundary

Applies to positive integers only; excludes perfect numbers (equality) and deficient numbers. The property depends on proper divisors only and is independent of factor ordering or prime representation except insofar as they determine divisors.

Semantic Tension

Tension exists with the class of weird numbers (abundant but not expressible as a sum of distinct proper divisors) and notions like multiply abundant; focus can be on numerical excess or on representability of that excess.

Synthesis

An abundant number is one whose proper divisors collectively outweigh it, marking it as imbalanced toward surplus and serving as a node in the taxonomy between perfect and deficient integers with further refinements like semiperfect or weird.