Deficient Numbers

Complete list of deficient numbers up to 200

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 51 52 53 55 57 58 59 61 62 63 64 65 67 68 69 71 73 74 75 76 77 79 81 82 83 85 86 87 89 91 92 93 94 95 97 98 99 101 103 105 106 107 109 110 111 113 115 116 117 118 119 121 122 123 124 125 127 128 129 130 131 133 134 135 136 137 139 141 142 143 145 146 147 148 149 151 152 153 154 155 157 158 159 161 163 164 165 166 167 169 170 171 172 173 175 177 178 179 181 182 183 184 185 187 188 189 190 191 193 194 195 197 199

What are Deficient Numbers?

A deficient number (also called a defective number) is a positive integer for which the sum of its proper divisors is less than the number itself. Most positive integers are deficient. For example, 8 has proper divisors 1, 2, and 4, and their sum is 1 + 2 + 4 = 7, which is less than 8. The deficiency of 8 is 8 − 7 = 1.

The first several deficient numbers are: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25...

Deficient Numbers and Primes

All prime numbers are deficient because the only proper divisor of any prime p is 1, and 1 < p for all primes. Similarly, all powers of primes (p², p³, etc.) are deficient. For example, 25 = 5² has proper divisors 1 and 5, summing to 6, which is far less than 25.

The most deficient numbers (those with the largest relative deficiency) are primes, where the sum of proper divisors is just 1. Among composite numbers, the deficiency varies widely. Numbers with few prime factors tend to be more deficient, while numbers with many small prime factors tend to be abundant.

Density and Distribution

Deficient numbers are the most common type in the classification of integers by divisor sums. Approximately 75.24% of all positive integers are deficient, about 24.76% are abundant, and perfect numbers have density zero. This means that roughly three out of every four positive integers are deficient.

Among the first 100 positive integers, 76 are deficient, 22 are abundant, and only 2 are perfect (6 and 28). The distribution is not uniform — deficient numbers are more concentrated among numbers with fewer divisors, while abundant numbers cluster around highly composite numbers.

Properties

Any divisor of a deficient number is also deficient. This is because removing a number's factors can only reduce the divisor sum relative to the number. All numbers of the form 2n are deficient, since their proper divisor sum is 2n − 1 (one less than the number). All numbers of the form paqb where p and q are large primes are deficient.

The concept of deficiency is measured as d(n) = 2n − σ(n), where σ(n) is the sum of all divisors. A number is deficient when d(n) > 0, perfect when d(n) = 0, and abundant when d(n) < 0. This measure provides a way to compare how "deficient" different numbers are.

Historical Context

The classification of numbers as deficient, perfect, or abundant dates back to ancient Greek mathematics. Nicomachus of Gerasa in his Introduction to Arithmetic (circa 100 AD) described deficient numbers metaphorically as those that fall short, like creatures with too few parts. This three-way classification remains fundamental in number theory to this day.