Abundancy index calculator

The abundancy index of a positive integer n is the exact rational I(n) = sigma(n) divided by n, where sigma(n) is the sum of every positive divisor of n including one and n itself. Number theorists use this ratio to classify n as deficient when I(n) is less than two, perfect when I(n) equals two, and abundant when I(n) exceeds two, and they compare indices when studying friendly numbers, solitary numbers, and aliquot structure. This abundancy index calculator accepts a positive integer n, computes sigma(n) from the prime factorization, reduces sigma(n)/n by their greatest common divisor, and returns the fraction in lowest terms. Values of n below one are rejected as invalid input.

What the abundancy index is and why the exact fraction matters

The abundancy index I(n) of a positive integer n is the rational number sigma(n) divided by n, where sigma(n) denotes the sum of all positive divisors of n, including both one and n. The definition is classical: Euler already used the ratio when studying perfect numbers, and modern papers write I(n) whenever they need to compare how densely the divisors of n fill the interval up to n.

Because sigma is a multiplicative function, I is multiplicative as well: if a and b are coprime then I(ab) equals I(a) times I(b). That identity is why a prime p has index (p+1)/p, a prime power p^k has index (p^{k+1}−1)/((p−1)p^k), and a square-free product of distinct primes has a product of those simple fractions. Comparing I(n) with the integer two is exactly the perfect-number trichotomy: I(n) less than two means n is deficient, I(n) equal to two means n is perfect, and I(n) greater than two means n is abundant.

Twelve is the smallest abundant integer because sigma(12) is twenty-eight, so I(12) equals twenty-eight over twelve, which reduces to seven thirds. Six is the smallest perfect integer because sigma(6) is twelve and the reduced index is two over one. Ten is deficient because sigma(10) is eighteen and eighteen over ten reduces to nine fifths, which sits strictly below two.

Returning the reduced fraction, not a floating approximation, is the whole point of this tool. Friendly-number searches ask whether I(a) equals I(b) as rationals, and a decimal such as 2.333 conceals that twenty-eight twelfths and seven thirds are the same class. One has index one over one and is deficient; that edge case is accepted, not rejected.

Input contract, exact algorithm, and the response you get back

Internally the engine factors n by trial division, walking even powers of two first and then odd trial primes up to the square root of the remaining cofactor. For each prime power p^a it multiplies the integer geometric series one plus p plus p squared up to p^a to accumulate sigma(n).

It then computes g equal to the greatest common divisor of sigma(n) and n with the Euclidean algorithm and returns numerator equal to sigma(n) divided by g and denominator equal to n divided by g, together with the display string numerator slash denominator. The response also echoes n, the integer sigma, and a kind label that is deficient, perfect, or abundant according to whether sigma is less than, equal to, or greater than twice n.

All arithmetic stays in exact integers; there is no Date, no random draw, and no network call. Pair this page with a sum-of-divisors tool only when you want sigma(n) alone.

Where an abundancy index calculator earns its keep

Students meet the abundancy index when a textbook first defines deficient, perfect, and abundant numbers, because the comparison of I(n) with two is easier to remember than the equivalent test on sigma(n) and twice n. Contest writers and learning-management authors need an oracle that returns the exact reduced fraction so a grader can accept seven thirds and twenty-eight twelfths as the same answer after reduction, and so a learner who writes a rounded decimal is marked incomplete rather than lucky.

Researchers who hunt friendly numbers, solitary numbers, or integers with a prescribed abundancy require exact rational equality, not a float epsilon that quietly identifies 7/3 with a nearby impostor. Classroom notebooks can tabulate I(n) against n for the first few hundred integers without installing a computer-algebra system. Script authors who walk aliquot sequences often log the index next to the aliquot sum so a spike above two is visible before the sequence diverges.