Number that is more than the sum of its proper divisors
Demonstration, withCuisenaire rods, of the deficiency of the number 8
Innumber theory, adeficient number ordefective number is apositive integern for which thesum of divisors ofn is less than2n. Equivalently, it is a number for which the sum ofproper divisors (oraliquot sum) is less thann. For example, the proper divisors of 8 are1, 2, and 4, and their sum is less than 8, so 8 is deficient.
Denoting byσ(n) the sum of divisors, the value2n −σ(n) is called the number'sdeficiency. In terms of the aliquot sums(n), the deficiency isn −s(n).
As an example, consider the number 21. Its proper divisors are 1, 3 and 7, and their sum is 11. Because 11 is less than 21, the number 21 is deficient. Its deficiency is 21 − 11 = 10.
Since the aliquot sums of prime numbers equal 1, allprime numbers are deficient.[1] More generally, all odd numbers with one or two distinct prime factors are deficient. It follows that there are infinitely manyodd deficient numbers. There are also an infinite number ofeven deficient numbers as allpowers of two have the sum (1 + 2 + 4 + 8 + ... + 2x − 1 = 2x − 1). The infinite family of numbers of form 2n − 1 ×pm wherem > 0 andp is a prime > 2n − 1 are also deficient.
More generally, allprime powers are deficient, because their only proper divisors are which sum to, which is at most.[2]
All properdivisors of deficient numbers are deficient.[3] Moreover, all proper divisors ofperfect numbers are deficient.[4]
There exists at least one deficient number in the interval for all sufficiently largen.[5]
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.JSTOR27958492.
