In number theory, a deficient number or defective number is a positive integern for which the sum of divisors of n is less than 2n. 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.
Demonstration, with Cuisenaire rods, of the deficiency of the number 8
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).
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.
Propertiesedit
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 + ... + 2x-1 = 2x - 1).
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]
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. ISBN1-4020-4215-9. Zbl 1151.11300.
deficient, number, number, theory, deficient, number, defective, number, positive, integer, which, divisors, less, than, equivalently, number, which, proper, divisors, aliquot, less, than, example, proper, divisors, their, less, than, deficient, demonstration,. 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 2n 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 Demonstration with Cuisenaire rods of the deficiency of the number 8 Denoting by s n the sum of divisors the value 2n s n is called the number s deficiency In terms of the aliquot sum s n the deficiency is n s n Contents 1 Examples 2 Properties 3 Related concepts 4 See also 5 Notes 6 References 7 External linksExamples editThe 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 editSince 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 2x 1 2x 1 More generally all prime powers p k displaystyle p k nbsp are deficient because their only proper divisors are 1 p p 2 p k 1 displaystyle 1 p p 2 dots p k 1 nbsp which sum to p k 1 p 1 displaystyle frac p k 1 p 1 nbsp which is at most p k 1 displaystyle p k 1 nbsp 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 displaystyle n n log n 2 nbsp for all sufficiently large n 5 Related concepts edit nbsp Euler diagram of numbers under 100 Abundant Primitive abundant Highly abundant Superabundant and highly composite Colossally abundant and superior highly composite Weird Perfect Composite Deficient Closely related to deficient numbers are perfect numbers with s n 2n and abundant numbers with s n gt 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 See also editAlmost perfect number Amicable number Sociable number Superabundant numberNotes edit 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 Sandor Mitrinovic amp Crstici 2006 p 108 Dickson 1919 p 3 References editDickson 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 Sandor Jozsef Mitrinovic Dragoslav S Crstici Borislav eds 2006 Handbook of number theory I Dordrecht Springer Verlag ISBN 1 4020 4215 9 Zbl 1151 11300 External links editThe Prime Glossary Deficient number Weisstein Eric W Deficient Number MathWorld deficient number at PlanetMath Retrieved from https en wikipedia org w index php title Deficient number amp oldid 1211620839, wikipedia, wiki, book, books, library,
