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Proper Divisors Calculator

Enter a whole number to list its proper divisors, count and aliquot sum.

How it is calculated

The tool tries every integer i from 1 while i × i ≤ n. Whenever n mod i = 0, both i and its partner n ÷ i are divisors, so only about the square root of n checks are needed. Any divisor equal to n itself is dropped, and a square root that pairs with itself is counted once. The remaining divisors are sorted, counted, and added up to give the aliquot sum: s(n) = σ(n) − n. The input must be a whole number from 1 to 10,000,000; decimals, zero, negatives and larger values give no result, and a thousands separator such as 1,000 is not accepted. For n = 1 the list is empty and shown as (none), with a count and sum of 0.

Worked example

Take n = 36. Testing i from 1 to 6 finds the pairs 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 alone because 6 × 6 = 36. Dropping 36 itself leaves the proper divisors 1, 2, 3, 4, 6, 9, 12, 18. That is a count of 8, and the aliquot sum is 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55. Since 55 is greater than 36, the number is abundant. Comparing the sum with n like this is how you can tell perfect, abundant and deficient numbers apart.

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Common mistakes

  • Counting the number itself as a divisor. A proper divisor list never includes n, so the count here is one less than the usual number of divisors of n.
  • Expecting 1 to be excluded. 1 is always a proper divisor of every n above 1, which is why a prime shows just 1 with a sum of 1.
  • Typing a number with a thousands separator, such as 1,000, or a decimal. Both are rejected and give no result.
  • Entering values above 10,000,000, which are outside the supported range.
  • Confusing the aliquot sum with the sum of all divisors. The aliquot sum leaves out n, so a perfect number like 28 has an aliquot sum of exactly 28.

About this calculator

A proper divisor of a number is any divisor other than the number itself — for 12, that's 1, 2, 3, 4, and 6. Adding proper divisors together gives the aliquot sum, a value ancient Greek mathematicians used to classify numbers: if the aliquot sum equals the number, it's a perfect number (like 6 = 1+2+3); if the sum is higher, it's abundant; if lower, it's deficient.

This calculator finds proper divisors by testing factor pairs up to the square root of the number, then reports the count and aliquot sum. Number theorists, math teachers, and puzzle enthusiasts use these ideas to study perfect and amicable numbers, and the same divisor-counting logic underlies parts of cryptography and computer science.

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Comments (1)

  • Editorial team

    Try feeding the aliquot sum back in as the next input. Starting at 220 the sum is 284, and 284 gives back 220, which is an amicable pair; 6 and 28 return themselves. Repeating this by hand traces an aliquot sequence, and the tool reports one step at a time. Have you found a number whose chain ends at 1 after only a few steps? Post the start value.

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