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Sumivo · Mathematics

Divisors Calculator

Find a whole number's divisors and its divisor functions — the count d(n), the sum σ(n), the aliquot sum, and whether the number is perfect, abundant, or deficient.

01 / inputs

integer

02 / result

CLASSIFICATION

Abundant

Divisors d(n)
6
Sum σ(n)
28
Aliquot sum
16
Divisors
1, 2, 3, 4, 6, 12

Calculation trace

  1. 6
    6
    Number of divisors d(n)
  2. 1 + 2 + 3 + 4 + 6 + 12 = 28
    28
    Sum of divisors σ(n)
  3. 28 − 12 = 16
    16
    Aliquot sum (σ(n) − n)
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How it works

A divisor of a whole number divides it exactly, leaving no remainder — for positive numbers the divisors are the same as the factors. The tool finds them all by trial up to the square root of n, then reports three number-theory quantities: d(n), the count of divisors; σ(n), the sum of all divisors; and the aliquot sum, σ(n) − n, which adds up only the proper divisors (everything except n itself). The aliquot sum decides the classification: a number is perfect when it equals its aliquot sum (6 = 1 + 2 + 3), abundant when the aliquot sum is larger (12, whose proper divisors sum to 16), and deficient when it is smaller (every prime is deficient, since its only proper divisor is 1).

Assumptions & limits

  • Defined for whole numbers of 1 or greater; zero, negatives, and decimals are rejected.
  • Counts positive divisors only; d(n) and σ(n) both include 1 and n.
  • Inputs are capped at 1,000,000,000,000 so the search stays fast in the browser.
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FAQ

What does d(n) mean?
d(n) is the number of positive divisors of n. For 12 the divisors are 1, 2, 3, 4, 6, 12, so d(12) = 6.
What does σ(n) mean?
σ(n) is the sum of all positive divisors of n, including n itself. For 12, σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.
What is a perfect number?
A number that equals the sum of its proper divisors (its aliquot sum). 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are the first two perfect numbers.
What is the difference between abundant and deficient numbers?
A number is abundant when its proper divisors sum to more than the number (12 → 16) and deficient when they sum to less (10 → 8). Perfect numbers sit exactly on the boundary.
How is this different from the factors calculator?
The factors calculator focuses on listing every factor and its factor pairs. This one focuses on the divisor functions — how many divisors there are, their sum, and whether the number is perfect, abundant, or deficient.
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