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Category Number Generators

How many abundant numbers? Between 1 and 10,000

Show divisors Odd abundant only

Presets:

#	Number	σ(n)	s(n)	Abundance	Divisors

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About

An abundant number (or excessive number) is a positive integer n for which the sum of its proper divisors σ(n) − n exceeds the number itself. The smallest abundant number is 12, whose proper divisors 1, 2, 3, 4, 6 sum to 16, yielding an abundance of 4. Roughly 24.8% of positive integers are abundant. Misidentifying number classification breaks algorithms in cryptography, coding theory, and combinatorial optimization that depend on divisor structure. Every integer greater than 20161 can be expressed as a sum of two abundant numbers (Schur's theorem, proved 1931). This generator computes the full divisor sum σ(n) via trial division up to √n, guaranteeing correctness. It does not use probabilistic shortcuts. Note: computation time scales with range size. Ranges above 1,000,000 may take several seconds.

Formulas

A positive integer n is classified by comparing the sum of all its positive divisors, denoted σ(n), against 2n. The divisor sum function is computed as:

σ(n) = ∑d | n d

The aliquot sum s(n) equals the sum of proper divisors only:

s(n) = σ(n) − n

The abundance of n is defined as:

A(n) = s(n) − n = σ(n) − 2n

Classification follows a piecewise rule:

{

Abundant if A(n) > 0Perfect if A(n) = 0Deficient if A(n) < 0

The algorithm iterates trial divisors from 1 to √n. For each d that divides n, both d and nd are accumulated (avoiding double-counting when d = √n). This yields O(√n) per number.

Where σ(n) = sum of all positive divisors of n, s(n) = aliquot sum (proper divisors only), A(n) = abundance (excess over n), d = a positive divisor of n.

Reference Data

Number	Proper Divisor Sum	Abundance	Divisors	Classification
12	16	4	1, 2, 3, 4, 6	Abundant
18	21	3	1, 2, 3, 6, 9	Abundant
20	22	2	1, 2, 4, 5, 10	Abundant
24	36	12	1, 2, 3, 4, 6, 8, 12	Abundant
30	42	12	1, 2, 3, 5, 6, 10, 15	Abundant
36	55	19	1, 2, 3, 4, 6, 9, 12, 18	Abundant
40	50	10	1, 2, 4, 5, 8, 10, 20	Abundant
42	54	12	1, 2, 3, 6, 7, 14, 21	Abundant
48	76	28	1, 2, 3, 4, 6, 8, 12, 16, 24	Abundant
54	66	12	1, 2, 3, 6, 9, 18, 27	Abundant
56	64	8	1, 2, 4, 7, 8, 14, 28	Abundant
60	108	48	1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30	Abundant
66	78	12	1, 2, 3, 6, 11, 22, 33	Abundant
70	74	4	1, 2, 5, 7, 10, 14, 35	Abundant
72	123	51	1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36	Abundant
78	90	12	1, 2, 3, 6, 13, 26, 39	Abundant
80	106	26	1, 2, 4, 5, 8, 10, 16, 20, 40	Abundant
84	140	56	1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42	Abundant
88	92	4	1, 2, 4, 8, 11, 22, 44	Abundant
90	144	54	1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45	Abundant
96	156	60	1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48	Abundant
100	117	17	1, 2, 4, 5, 10, 20, 25, 50	Abundant
102	114	12	1, 2, 3, 6, 17, 34, 51	Abundant
104	106	2	1, 2, 4, 8, 13, 26, 52	Abundant
108	172	64	1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54	Abundant

Frequently Asked Questions

The proper divisors of 12 are 1, 2, 3, 4, 6, summing to 16. Since 16 > 12, the number is abundant with abundance 4. Every integer below 12 is either deficient or perfect (6 is the first perfect number). No odd abundant number exists below 945.

No. The even number 2 has only the proper divisor 1, making it deficient. However, every multiple of 6 greater than 6 is abundant. The density of abundant numbers among all positive integers converges to approximately 24.8% (Davenport, 1933). Even abundant numbers dominate: the first odd abundant number is 945.

Perfect numbers satisfy s(n) = n exactly, meaning abundance A(n) = 0. They are the boundary between deficient and abundant. Only 51 even perfect numbers are known (as of 2024), all of the Euclid-Euler form 2^p−1(2^p − 1). No odd perfect numbers have been found.

Each candidate n requires O(√n) operations for trial division. For a range [1, N], total work is approximately O(N^3/2). Ranges up to 100,000 complete in under one second. Ranges up to 1,000,000 may take 2 - 5 seconds. The tool offloads computation to a Web Worker to prevent UI freezing.

Yes. The first pair of consecutive abundant numbers is (5775, 5776). Consecutive triples also exist. Erdős proved that the set of abundant numbers has a Schnirelmann density, implying arbitrarily long runs of consecutive abundant numbers exist, though they become increasingly rare at small magnitudes.

Divisor structure analysis appears in cryptographic key generation (selecting primes with specific divisor properties), error-correcting codes (perfect codes relate to perfect numbers), and scheduling theory (highly composite numbers, a subset of abundant numbers, optimize divisor counts for task allocation). The aliquot sum also drives aliquot sequence analysis, which connects to unsolved problems like the Catalan-Dickson conjecture.