number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, a deficient number or defective number is a

positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

for which the sum of divisors of is less than . Equivalently, it is a number for which the sum of proper divisors (or

aliquot sum In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. That is, s(n)=\sum_ d \, . It can be used to characterize the prime numbers, perfect numbers, sociabl ...

) is less than . For example, the proper divisors of 8 are , and their sum is less than 8, so 8 is deficient. Denoting by the sum of divisors, the value is called the number's deficiency. In terms of the aliquot sum , the deficiency is .

Examples

The first few deficient numbers are :1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ... As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 42, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.

Properties

Since the aliquot sums of prime numbers equal 1, all

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s are deficient. More generally, all odd numbers with one or two distinct prime factors are deficient. It follows that there are infinitely many odd deficient numbers. There are also an infinite number of even deficient numbers as all powers of two have the sum (). The infinite family of numbers of form 2^(n - 1) * p^m where m > 0 and p is a prime > 2^n - 1 are also deficient. More generally, all prime powers

p^k

are deficient, because their only proper divisors are

1, p, p^2, \dots, p^

which sum to

\frac

, which is at most

p^k-1

. All proper divisors of deficient numbers are deficient. Moreover, all proper divisors of

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...

s are deficient. There exists at least one deficient number in the interval

, n + (\log n)^2 /math> for all sufficiently large ''n''.

Related concepts

Closely related to deficient numbers are

s with ''σ''(''n'') = 2''n'', and

abundant number In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total ...

s with ''σ''(''n'') > 2''n''. Nicomachus was the first to subdivide numbers into deficient, perfect, or abundant, in his ''

Introduction to Arithmetic Nicomachus of Gerasa (; ) was an Ancient Greek Neopythagoreanism, Neopythagorean philosopher from Gerasa, in the Syria (Roman province), Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical pr ...

'' (circa 100 CE). However, he applied this classification only to the even numbers.

Notes

References

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External links