Deficient number

In number theory, a deficient number or defective number is a positive integer $n$ for which the sum of divisors of $n$ is less than $2 n$ . Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than $n$ . For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient.

Denoting by $σ (n)$ the sum of divisors, the value $2 n - σ (n)$ is called the number's deficiency. In terms of the aliquot sum $s (n)$ , the deficiency is $n - s (n)$ .

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, ... (sequence A005100 in the OEIS)

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 numbers are deficient.^[1] 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 ( $1 + 2 + 4 + 8 + ... + 2 x -1 = 2 x - 1$ ).

More generally, all prime powers $p^{k}$ are deficient, because their only proper divisors are $1,p,p^{2},\dots ,p^{k-1}$ which sum to ${\frac {p^{k}-1}{p-1}}$ , which is at most $p^{k}-1$ .^[2]

All proper divisors of deficient numbers are deficient.^[3] Moreover, all proper divisors of perfect numbers are deficient.^[4]

There exists at least one deficient number in the interval $[n,n+(\log n)^{2}]$ for all sufficiently large n.^[5]

Related concepts

Closely related to deficient numbers are perfect numbers with σ(n) = 2n, and abundant numbers with σ(n) > 2n.

Nicomachus was the first to subdivide numbers into deficient, perfect, or abundant, in his Introduction to Arithmetic (circa 100 CE). However, he applied this classification only to the even numbers.^[6]

Notes

^ Prielipp (1970), Theorem 1, pp. 693–694.
^ Prielipp (1970), Theorem 2, p. 694.
^ Prielipp (1970), Theorem 7, p. 695.
^ Prielipp (1970), Theorem 3, p. 694.
^ Sándor, Mitrinović & Crstici (2006), p. 108.
^ Dickson (1919), p. 3.

References

Dickson, Leonard Eugene (1919). History of the Theory of Numbers, Vol. I: Divisibility and Primality. Carnegie Institute of Washington.
Prielipp, Robert W. (1970). "Perfect numbers, abundant numbers, and deficient numbers". The Mathematics Teacher. 63 (8): 692–696. doi:10.5951/MT.63.8.0692. JSTOR 27958492.
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.