Perfect Number Calculator - Sum of Divisors = 2n

Our perfect number checker tests any positive integer up to 100,000,000. Enter a number in the input field, click Check, and the calculator instantly:

1. Finds all proper divisors (factors of n smaller than n)
2. Computes the aliquot sum — the sum of those proper divisors
3. Compares the aliquot sum with n to classify it as perfect, abundant, or deficient
4. Displays the abundancy index σ(n)/n and full divisor analysis

A number is perfect when its aliquot sum equals the number, abundant when the sum is greater, and deficient when the sum is less. For example, 12 is abundant (1+2+3+4+6 = 16 > 12), while 10 is deficient (1+2+5 = 8 < 10).

Every even perfect number can be generated with the Euler–Euclid formula: if 2^p − 1 is a Mersenne prime, then 2^p−1 × (2^p − 1) is a perfect number. Our perfect number generator lets you pick a Mersenne exponent and instantly produce the corresponding perfect number.

The fifth perfect number is 33,550,336, discovered in 1461 by an unknown mathematician. Notice how rapidly perfect numbers grow — the 6th is already 8,589,869,056, and the largest known perfect number has over 49 million digits.

The aliquot sum s(n) of a positive integer n is the sum of all its proper divisors — every positive factor of n except n itself. Our calculator displays the aliquot sum for every number you check.

• If s(n) = n, the number is perfect (e.g., s(28) = 1+2+4+7+14 = 28)
• If s(n) > n, the number is abundant (e.g., s(12) = 1+2+3+4+6 = 16)
• If s(n) < n, the number is deficient (e.g., s(10) = 1+2+5 = 8)

The aliquot sum is central to number theory. It defines amicable numbers (pairs where s(a) = b and s(b) = a), sociable numbers (chains of aliquot sums), and the open Catalan–Dickson conjecture about aliquot sequences. Use our calculator alongside the factor calculator or divisibility calculator for deeper divisor exploration.

Every even perfect number is tied to a Mersenne prime — a prime of the form 2^p − 1. Euler proved the converse of Euclid's theorem: every even perfect number has the form 2^p−1(2^p − 1). This tight connection means that discovering a new Mersenne prime automatically yields a new perfect number.

As of 2024, 51 Mersenne primes are known (via the GIMPS project), so 51 even perfect numbers are known. The largest has over 49 million digits. Whether any odd perfect number exists remains unsolved — if one exists, researchers have shown it must exceed 10²⁰⁰⁰ and have at least 101 prime factors.

To explore the prime foundations of these numbers, try our prime calculator and prime factorization calculator. For computing the large powers of 2 in the Euler–Euclid formula, our exponent calculator is a helpful companion tool.

Perfect numbers exhibit surprising mathematical patterns that connect them to many areas of number theory:

• Triangular numbers: Every even perfect number is also a triangular number. For instance, 28 = 1+2+3+4+5+6+7, and 6 = 1+2+3.
• Digital root: Except for 6, every even perfect number has a digital root of 1 and ends in either 6 or 8 in decimal.
• Reciprocal sum: The sum of the reciprocals of all divisors of a perfect number always equals 2. For 6: 1/1 + 1/2 + 1/3 + 1/6 = 2.
• Binary representation: In binary, even perfect numbers look elegant — for example, 6 = 110₂, 28 = 11100₂, 496 = 111110000₂.
• Abundancy index: A number is perfect if and only if its abundancy index σ(n)/n = 2. Our calculator displays this metric for every input.

These properties make perfect numbers a natural bridge between elementary arithmetic and advanced number theory. Students can verify each property using our calculator alongside the greatest common factor calculator and even odd calculator.

This tool is designed for students, teachers, and math enthusiasts exploring number theory. Here are practical ways to use it:

For related number theory exploration, our composite number calculator helps classify numbers by their prime structure, and the arithmetic sequence calculator reveals patterns in mathematical sequences.