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Abundant number

An abundant number is a positive integer n for which the sum of its proper divisors exceeds n itself, where proper divisors are the positive divisors excluding n.^[1] Equivalently, if \sigma(n) denotes the sum of all positive divisors of n, then n is abundant when \sigma(n) > 2n.^[1] The abundance of such a number is defined as \sigma(n) - 2n, a positive value distinguishing it from perfect numbers (where equality holds) and deficient numbers (where the sum is less).^[1] The concept of abundant numbers originated in ancient Greek mathematics, with the term introduced by Nicomachus of Gerasa around 100 CE in his work Introductio Arithmetica, where he classified numbers based on the relationship between a number and the sum of its proper divisors.^[2] The smallest abundant number is 12, whose proper divisors (1, 2, 3, 4, and 6) sum to 16; subsequent examples include 18, 20, 24, and 30.^[1] All abundant numbers below 100 are even, and the smallest odd abundant number is 945 ($3^3 \times 5 \times 7).^[1] A key property is that every multiple of an abundant number is itself abundant, contributing to their distribution among integers.^[1] Abundant numbers exhibit interesting asymptotic behavior, with their natural density estimated to lie between 0.247619608 and 0.247619658 as of 2025.^[3] Primitive abundant numbers, defined as those whose proper divisors are all deficient, include 945 as the smallest example.^[1] Notably, every integer greater than 20161 can be expressed as the sum of two abundant numbers, a result analogous to Goldbach's conjecture for even numbers.^[1] These properties have been studied extensively since the early 20th century by mathematicians such as Davenport and Erdős, highlighting the role of abundant numbers in number theory and divisor functions.^[1]

Fundamentals

Definition

In number theory, the sum-of-divisors function \sigma(n) denotes the sum of all positive divisors of a natural number n, including 1 and n itself. The sum of proper divisors, denoted s(n), is defined as \sigma(n) - n, consisting of the aliquot parts of n—that is, its positive divisors excluding n.^[4]^[5]^[6] A natural number n is called abundant if s(n) > n, or equivalently, if \sigma(n) > 2n. The abundance of such an n is measured by \sigma(n) - 2n (or s(n) - n), which is strictly positive. This classification of numbers based on the relationship between n and the sum of its proper divisors was first introduced by the ancient Greek mathematician Nicomachus around 100 AD.^[1]^[7]^[8]^[9] For example, consider n = 12. Its positive divisors are 1, 2, 3, 4, 6, and 12, so \sigma(12) = 28 and s(12) = 16. Since $16 > 12 (or $28 > 24), 12 is abundant with abundance $28 - 24 = 4.^[1]

Historical Background

The concept of abundant numbers traces its origins to the ancient Greek mathematician Nicomachus of Gerasa, who around 100 AD introduced the classification of positive integers based on the relationship between a number and the sum of its proper divisors in his work Introductio Arithmetica. Nicomachus described numbers whose proper divisors sum to more than the number itself as "superabundant," contrasting them with "perfect" numbers (where the sum equals the number) and "deficient" numbers (where the sum is less). He illustrated superabundant numbers with analogies to excess in nature, such as animals with superfluous features, emphasizing their philosophical and moral implications within Pythagorean arithmetic traditions.^[10]^[9] In the ancient Greek context, this classification formed part of broader studies in arithmetic that viewed numbers as embodying harmony or imbalance, with abundant numbers representing excess in the cosmic order. The ideas persisted into the medieval period, where Islamic mathematicians like Ibn Sina explored abundant numbers alongside related concepts such as perfect and amicable numbers in treatises on divisor sums, though without significant new advancements on abundance specifically.^[11] During the Renaissance, European scholars, including Leonardo of Pisa (Fibonacci), revived interest through works like Liber Abaci (1202), where he detailed methods for generating even perfect numbers using the Euclidean formula, thereby sustaining the Nicomachean framework amid limited further development on abundant classifications.^[12] The modern study of abundant numbers revived in the early 20th century, driven by historical compilations and emerging computational tools in number theory. Leonard Eugene Dickson's comprehensive History of the Theory of Numbers (1919–1923) synthesized early references to abundant and related numbers, providing a foundational timeline that highlighted their scarcity in pre-modern literature. From the 1930s onward, interest grew with analytical work on their distribution; for instance, Felix Behrend's 1932–1933 papers established initial bounds on the density of abundant numbers among positives integers, marking the shift toward rigorous probabilistic analysis. Subsequent decades saw continued exploration through computational verification, though no major conceptual breakthroughs emerged by 2025.^[13]

Examples

Small Abundant Numbers

The smallest abundant number is 12, whose proper divisors (1, 2, 3, 4, 6) sum to 16, exceeding 12.^[1] This is followed by 18, with proper divisors (1, 2, 3, 6, 9) summing to 21 > 18; 20, with proper divisors (1, 2, 4, 5, 10) summing to 22 > 20; 24, with proper divisors (1, 2, 3, 4, 6, 8, 12) summing to 36 > 24; and 30, with proper divisors (1, 2, 3, 5, 6, 10, 15) summing to 42 > 30.^[1]^[14] The sequence of the first 20 abundant numbers is 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90.^[14] All abundant numbers less than 100 are even, illustrating that small examples of this concept occur exclusively among even integers.^[1] Many of these small abundant numbers, such as 12, 24, 36, 48, and 60, are also highly composite, meaning they set records for the number of divisors among smaller positive integers.^[15] The full sequence is cataloged in the Online Encyclopedia of Integer Sequences as A005101.^[14]

Odd Abundant Numbers

Odd abundant numbers are significantly rarer than their even counterparts due to the absence of the prime factor 2, which efficiently boosts the divisor sum relative to the number itself. The smallest such number is 945, whose proper divisors sum to 975, exceeding 945.^[1] This makes 945 the 232nd abundant number overall.^[16] The prime factorization of 945 is $3^3 \times 5 \times 7, demonstrating the necessity for a high multiplicity of the smallest odd prime (3) combined with the next two small odd primes to achieve abundance without even factors.^[1] The subsequent odd abundant numbers include 1575 ($3^2 \times 5^2 \times 7), 2205 ($3^2 \times 5 \times 7^2), 2835 ($3^4 \times 5 \times 7), and 3465 ($3^2 \times 5 \times 7 \times 11), all sharing similar structures reliant on small odd primes.^[16] The scarcity of odd abundant numbers arises because, for an odd n, the abundancy index \sigma(n)/n must exceed 2 solely through contributions from odd primes, requiring either elevated powers or a greater number of distinct small odd prime factors to compensate for the lack of 2's multiplicative efficiency in the divisor function.^[1] Unlike even abundants, which can achieve this with fewer factors, odd examples demand more intricate factorizations to surpass the threshold. A notable computational advancement identifies the smallest odd abundant number not divisible by 3 or 5—thus with smallest prime factor 7—as $7^2 \times 11^2 \times [13](/page/13) \times 17 \times 19 \times 23 \times 29 \times 31 \times 37 \times 41 \times [43](/page/43) \times 47 \times [53](/page/53) \times [59](/page/59) \times [61](/page/61*) \times 67, a 26-digit integer approximately $2.01 \times 10^{25}.^[17] This result underscores the exponential growth in size as smaller odd primes are excluded, highlighting the delicate balance needed for odd abundance.

Properties

General Properties

An abundant number n satisfies \sigma(n) - n > n, where \sigma(n) denotes the sum of all positive divisors of n. A key property is that the set of abundant numbers is closed under multiplication by integers greater than or equal to 2: if m is abundant and k \geq 2, then k \cdot m is also abundant, since the divisor sum of the product exceeds twice the product by at least as much as for m itself, amplified by k.^[1]^[18] For example, 12 is abundant with proper divisors summing to 16, and its multiple 24 has proper divisors summing to 36.^[1] This closure extends to multiples of perfect numbers: if p is perfect (\sigma(p) = 2p) and k \geq 2, then k \cdot p is abundant, as the divisor sum grows beyond twice the product.^[1] Consequently, since 12 is abundant, all multiples of 12 are abundant.^[19] Abundant numbers cannot be prime, as the sum of proper divisors of a prime q is 1, which is less than q.^[1] Weird numbers provide an interesting subclass: they are abundant but not semiperfect, meaning no subset of their proper divisors sums exactly to the number itself. The smallest weird number is 70, with proper divisors 1, 2, 5, 7, 10, 14, 35 summing to 74 (>70), yet no combination yields 70.^[20]

Density and Distribution

The natural density of the set of abundant numbers among the positive integers lies between 0.247619608 and 0.247619658.^[3] This bound improves upon earlier estimates, such as 0.2474 to 0.2480 established through extensive computations counting the proportion of abundant numbers up to limits as high as $10^{10}.^[21] More refined estimates from 2010 had placed the asymptotic proportion between 0.2476171 and 0.2476475.^[13] A notable aspect of the distribution involves representations as sums of abundant numbers. Every integer greater than 20161 can be expressed as the sum of two abundant numbers.^[22] This result extends a related finding that every even integer greater than 46 is likewise a sum of two abundant numbers, with 46 being the largest even number not expressible in this form.^[22] Computational approaches to determining the density rely on efficient algorithms for evaluating the sum-of-divisors function \sigma(n). Techniques such as segmented sieves enable the rapid computation of \sigma(n) over large intervals, allowing counts of abundant numbers up to $10^9 or beyond by identifying multiples of small abundant numbers and verifying borderline cases.^[21] These methods have been iteratively improved, with recent advancements in 2025 providing the current tight bounds through enhanced computational techniques.^[3]

Special Types

Primitive Abundant Numbers

A primitive abundant number is an abundant number whose proper divisors are all non-abundant (i.e., deficient or perfect), meaning no proper divisor is abundant.^[23] This distinguishes them as the "minimal" generators within the set of abundant numbers, as every abundant number is either primitive or a multiple of a smaller primitive abundant number. The concept was discussed in early number theory literature, emphasizing their role in the structure of non-deficient numbers.^[24] The smallest primitive abundant number is 12, with prime factorization $2^2 \times 3, where its proper divisors (1, 2, 3, 4, 6) are all deficient or perfect but none abundant.^[23] The sequence of primitive abundant numbers begins 12, 18, 20, 30, 42, 56, 66, 70, 78, 88, and continues infinitely, including both even and odd terms (OEIS A091191).^[23] The smallest odd primitive abundant number is 945 ($3^3 \times 5 \times 7), which is also the smallest odd abundant number overall; the sequence of odd primitive abundants starts 945, 1575, 2205, ... (OEIS A006038). Primitive abundant numbers typically feature small prime factors raised to relatively high powers to achieve abundance while keeping all proper substructures deficient, as higher primes would insufficiently boost the divisor sum relative to the number's size.^[25] For instance, 945's structure with the smallest odd primes and an elevated power on 3 ensures \sigma(945) = 1925 > 1890 = 2 \times 945, yet divisors like 315 ($3^2 \times 5 \times 7) yield \sigma(315) = 624 < 630. Identifying such numbers computationally involves calculating the sum-of-divisors function \sigma for the candidate and every proper divisor, a process that scales poorly for large candidates due to the need for complete divisor factorization and summation.^[26]

Highly Abundant Numbers

A highly abundant number is a positive integer n such that \sigma(n) > \sigma(k) for every positive integer k < n, where \sigma denotes the sum-of-divisors function.^[27] These numbers set successive records for the largest value attained by \sigma up to that point.^[28] Although the initial highly abundant numbers include deficient numbers (such as 1, 2, 3, 4, 8, and 10) and the perfect number 6, all highly abundant numbers starting from 12 are abundant, reflecting their increasingly rich divisor structures that push \sigma(n) well beyond $2n.^[28] Representative examples include 12 with \sigma(12) = 28, 60 with \sigma(60) = 168, 120 with \sigma(120) = 360, and 180 with \sigma(180) = 546.^[28] Highly abundant numbers often coincide with or resemble highly composite numbers, featuring exponents in their prime factorizations that decrease and incorporate small primes to maximize \sigma(n).^[27] They play a key role in analyzing the growth behavior of the divisor sum function, providing benchmarks for how \sigma(n) accumulates with additional prime factors.^[27] In contrast to superabundant numbers, which maximize the ratio \sigma(n)/n relative to all smaller integers, highly abundant numbers prioritize the absolute magnitude of \sigma(n).^[27] Every superabundant number is highly abundant, though it remains unknown whether infinitely many highly abundant numbers fail to be superabundant.^[27] The count of highly abundant numbers less than x satisfies a lower bound of (1 - \epsilon)(\log x)^2 for any \epsilon > 0 and sufficiently large x, alongside an upper bound of (\log x)^{c \log \log x} for some constant c > 0, underscoring their zero natural density within the positive integers.^[27]

Perfect and Deficient Numbers

Positive integers are classified into three categories based on the relationship between a number and the sum of its proper divisors, a trinary system introduced by the ancient Greek mathematician Nicomachus in his Introduction to Arithmetic around 100 AD.^[29] In this framework, abundant numbers exhibit an excess where the sum of proper divisors s(n) satisfies s(n) > n, perfect numbers achieve balance with s(n) = n, and deficient numbers show a deficit with s(n) < n.^[29] Perfect numbers are rare and well-studied, with the smallest examples being 6 (proper divisors: 1, 2, 3; sum = 6) and 28 (proper divisors: 1, 2, 4, 7, 14; sum = 28).^[30] All known perfect numbers are even, and by the Euclid-Euler theorem, every even perfect number takes the form $2^{p-1}(2^p - 1), where $2^p - 1 is a Mersenne prime (with p prime).^[30] No odd perfect numbers are known, and their existence remains an open problem.^[30] Deficient numbers form the majority of positive integers, comprising approximately 75.2% of them according to natural density estimates.^[21] Primes and powers of 2 are classic examples of deficient numbers, as their proper divisors sum to far less than the number itself. These categories are mutually exclusive: no number can be both abundant and perfect, as s(n) > n and s(n) = n cannot hold simultaneously.^[1] However, any multiple of a perfect number greater than the perfect number itself is abundant, since the divisor sum grows multiplicatively to exceed twice the value.^[1] Quasiperfect numbers, defined by s(n) = n + 1 (or equivalently, the total divisor sum \sigma(n) = 2n + 1), represent a slight excess beyond perfection but remain hypothetical, with no examples known despite extensive searches.^[31] Their existence is unsolved, though any such number must be odd, a square, and greater than $10^{35} with at least seven distinct prime factors.^[32]

Abundancy Index

The abundancy index of a positive integer n, denoted I(n), is defined as the ratio I(n) = \frac{\sigma(n)}{n}, where \sigma(n) is the sum of all positive divisors of n.^[33] This measure normalizes the divisor sum relative to the number itself, providing a scale-independent way to assess how "abundant" a number is in terms of its divisors. A number n is classified as abundant if I(n) > 2, perfect if I(n) = 2, and deficient if I(n) < 2.^[34] For instance, the smallest abundant number 12 has \sigma(12) = 28, yielding I(12) = \frac{28}{12} \approx 2.333, while the even perfect number 6 satisfies \sigma(6) = 12 and I(6) = 2.^[33] The abundancy index facilitates comparisons across numbers of varying sizes; for example, it reveals that multiples of perfect numbers have indices greater than 2, confirming their abundance. One key application of the abundancy index is in identifying friendly and solitary numbers. Friendly numbers are distinct positive integers m and n that share the same index, I(m) = I(n); for example, 6 and 28 are both perfect with I(6) = I(28) = 2, though more general friendly pairs exist among abundants.^[35] In contrast, solitary numbers have a unique abundancy index not matched by any other positive integer.^[36] This classification highlights structural similarities in divisor distributions beyond mere abundance thresholds. The abundancy index also connects to other arithmetic means of divisors. Specifically, the reciprocal $1/I(n) relates to the harmonic mean H(n) of the divisors of n, given by H(n) = \frac{\tau(n)}{I(n)}, where \tau(n) counts the divisors; thus, higher divisor richness corresponds to a lower relative harmonic mean.^[37] Regarding growth, for highly composite numbers, I(n) increases asymptotically like \log \log n, with the precise bound \limsup_{n \to \infty} \frac{I(n)}{\log \log n} = e^\gamma, where \gamma \approx 0.57721 is the Euler-Mascheroni constant, as established by Gronwall's theorem.

References

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Abundant Number -- from Wolfram MathWorld
An abundant number, sometimes also called an excessive number, is a positive integer n for which s(n)=sigma(n)-n>n, where sigma(n) is the divisor function.
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[PDF] A Study of the Sum of Divisors - Scholars' Mine
Some examples are as follows in which all numbers obtained are odd abundant numbers. The operation 5355 + 630 = 5985, but includes: 5355 + 420 = 5775 and ...
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Divisor Function -- from Wolfram MathWorld
The divisor function sigma_k(n) for n an integer is defined as the sum of the kth powers of the (positive integer) divisors of n, sigma_k(n)=sum_(d|n)d^k.
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Proper Divisor -- from Wolfram MathWorld
A positive proper divisor is a positive divisor of a number n, excluding n itself. For example, 1, 2, and 3 are positive proper divisors of 6, ...
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Aliquot Divisor -- from Wolfram MathWorld
The term "aliquot" is also frequently used to specifically mean a proper divisor, i.e., a divisor of a number other than the number itself. For example, the ...
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A005100 - OEIS
A number k is abundant if sigma(k) > 2k (cf. A005101), perfect if sigma(k) = 2k (cf. A000396), or deficient if sigma(k) < 2k (this sequence).
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The abundance of a number n , sometimes also called the abundancy (a term which in this work, is reserved for a different but related quantity), is the quantity ...Missing: definition | Show results with:definition
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Nicomachus divides numbers into three classes: the superabundant numbers ... Now satisfied with the moral considerations of numbers, Nicomachus goes on to ...
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### Extracted Sequence and Summary
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Highly composite numbers are numbers such that divisor function d(n)=sigma_0(n) (i.e., the number of divisors of n) is greater than for any smaller n.
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A "weird number" is a number that is abundant (ie, the sum of proper divisors is greater than the number) without being pseudoperfect.
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Abundant number - Wikipedia
Examples. The first 28 abundant numbers are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, ...Colossally abundant number · Highly abundant · Superabundant number
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Abundancy -- from Wolfram MathWorld
The abundancy of a number n is defined as the ratio sigma(n)/n, where sigma(n) is the divisor function.
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Friendly Pair -- from Wolfram MathWorld
A pair of distinct numbers (k,m) is a friendly pair (and k is said to be a friend of m ) if their abundancies are equal: Sigma(k)=Sigma(m).
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A074902 - OEIS
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