Enter a positive integer to classify it as deficient, perfect, or abundant
Enter Number
Result
Classification
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s(n) = sum of proper divisors
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Abundance = s(n) - n
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Detailed Derivation
Classification Criteria
Deficient: s(n) < n (most numbers)
Perfect: s(n) = n (e.g., 6, 28, 496)
Abundant: s(n) > n (e.g., 12, 18, 20, 24)
Abundance = s(n) - n
Every positive integer can be classified as deficient, perfect, or abundant based on whether the sum of its proper divisors (excluding itself) is less than, equal to, or greater than the number itself.
⚠All prime numbers are deficient (s(p)=1). The smallest abundant number is 12. The smallest odd abundant number is 945.
What Are Deficient and Abundant Numbers?
Number classification based on divisor sums has been studied since ancient Greek mathematics. Euclid and Nicomachus classified numbers into these three categories based on their relationship to the sum of divisors.
Deficient
s(n) < n. All primes, prime powers, and most even numbers. About 75% of numbers are deficient.
Abundant
s(n) > n. ~25% of numbers. All even numbers > 46 are abundant. Multiples of abundant numbers are abundant.
Primitive Abundant
Abundant numbers that are not multiples of a smaller abundant number: 12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102...
Greek Classification
Nicomachus compared numbers to living beings: deficient (deficient in self), perfect (self-sufficient), abundant (excessive).
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