Divisors

Divisors calculator

All the divisors of a number, how many there are and what they add up to, and whether the number is perfect, abundant or deficient.

Divisors6

Sum σ(n)56

Typeperfect

Its proper divisors (all except 28 itself) add up to 28, and 28 = 28.

All 6 divisors

Divisor pairs

1 × 28
2 × 14
4 × 7

From the prime factorization

28 = 2² × 7

Count: τ(n) = (2 + 1) × (1 + 1) = 6
Sum: σ(n) = (2³ − 1)/(2 − 1) × (1 + 7) = 56

Divisors from prime factors

A divisor of n is a whole number that divides it with nothing left over; 12 has six: 1, 2, 3, 4, 6 and 12. Divisors come in pairs, d × (n ÷ d) = n, so a hand search only needs to go up to √n. Perfect squares are the only numbers with an odd count, because their square root pairs with itself.

For bigger numbers the calculator starts from the prime factorization. If n = p^a × q^b × …, every divisor chooses an exponent from 0 to a for p, from 0 to b for q, and so on. That gives the count and the sum directly:

τ(n) = (a + 1)(b + 1)… σ(n) = (1 + p + … + p^a)(1 + q + … + q^b)…

For 360 = 2³ × 3² × 5 the count is 4 × 3 × 2 = 24 divisors and the sum is 15 × 13 × 6 = 1,170.

Now add the divisors except n itself and compare with n. Equal makes n perfect (6 = 1 + 2 + 3). More makes it abundant (for 12, 1 + 2 + 3 + 4 + 6 = 16). Less makes it deficient, as every prime is, since its only smaller divisor is 1.

Sources

OEIS sequences A000005 (number of divisors), A000203 (sum of divisors), A000396 (perfect numbers) and A005101 (abundant numbers).
Euclid, Elements, Book IX, Proposition 36: 2^p−1(2^p − 1) is perfect when 2^p − 1 is prime; Euler later showed every even perfect number has this form.
Ramanujan S. Highly composite numbers. Proceedings of the London Mathematical Society 14 (1915), 347–409.

Questions

Are there odd perfect numbers?

Nobody knows. Every known perfect number is even and of the form 2^p−1(2^p − 1) with 2^p − 1 prime, so there are exactly as many known perfect numbers as known Mersenne primes.

What is the smallest odd abundant number?

945 = 3³ × 5 × 7, whose proper divisors add up to 975. Even abundant numbers are common, since every multiple of 6 above 6 is abundant; odd ones are much rarer.

Which number below 1,000 has the most divisors?

840, with 32. Numbers that have more divisors than every smaller number are called highly composite; Ramanujan studied them in 1915.