Deficient number
In
Denoting by σ(n) the sum of divisors, the value 2n – 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
More generally, all prime powers are deficient, because their only proper divisors are which sum to , which is at most .[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 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.
See also
- Almost perfect number
- Amicable number
- Sociable number
- Superabundant number
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. JSTOR 27958492.
- Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Zbl 1151.11300.