HesapHanem

Deficient Number Checker

Enter a whole number to test whether it is deficient.

How it is calculated

The tool adds up all proper divisors of n (every divisor except n itself) and compares the sum s with n. abundance = s − n. If s < n the number is Deficient; if s = n (perfect) or s > n (abundant) the verdict is "Not deficient (perfect/abundant)". Divisors are found by trial division up to the square root of n, adding each pair i and n ÷ i (a repeated root in a square is counted once). Input must be a whole number of 1 or more; decimals and zero or negatives return no result. The number 1 has no proper divisors, so its sum is 0 and it counts as deficient. Note the sign: the abundance is negative for deficient numbers.

Worked example

Take n = 15. Its divisors are 1, 3, 5 and 15; leaving out 15 itself, the proper divisors are 1, 3 and 5. Their sum is 1 + 3 + 5 = 9. Since 9 is less than 15, the verdict is Deficient. Abundance = 9 − 15 = −6, so the sum falls 6 short of the number. By contrast 12 has proper divisors adding up to 16 and is not deficient.

Try this example →

Common mistakes

  • Including the number itself in the divisor sum. That sum (σ) is always above n for n above 1; the test uses the sum without n.
  • Expecting a positive deficiency. The tool shows abundance = sum − n, so a deficient number gives a negative figure (15 gives −6).
  • Reading "Not deficient" as "abundant". A result of 0 means a perfect number such as 6 or 28; only a positive figure means abundant.
  • Assuming odd numbers are always deficient. Odd abundant numbers exist, the smallest being 945 (sum 975).
  • Entering a decimal or negative value; only whole numbers from 1 upward are accepted.

About this calculator

A whole number is deficient when the sum of its proper divisors — all divisors excluding the number itself — is less than the number. For example, 8's proper divisors are 1, 2, and 4, which sum to 7, less than 8, making it deficient; the amount by which the number exceeds that sum (1, here) is called its deficiency. The checker shows the divisor sum and the sum minus the number, which is therefore a negative value for a deficient number (−1 for 8).

This classification, going back to ancient Greek number theory (Euclid and Nicomachus), splits all positive integers into three camps: deficient (divisor sum less than n), perfect (divisor sum exactly n, like 6 = 1+2+3 or 28), and abundant (divisor sum greater than n, like 12 = 1+2+3+4+6 = 16). Most numbers are actually deficient — all prime numbers are, since their only proper divisor is 1 — which is why perfect numbers are famously rare (only 52 are known as of the largest Mersenne-prime searches) and abundant numbers, while more common than perfect ones, are still a minority among the integers.

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Comments (1)

  • Editorial team

    The verdict lumps perfect and abundant numbers under one label, so read the third line: abundance is the divisor sum minus n. Zero means perfect (6, 28), positive means abundant (12 gives 4), negative means deficient (a prime p shows 1 minus p). Try 945, the smallest odd abundant number: it shows 30. Which number sent you here?

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