000RM.dvi

630 Perfect numbers

24.3 Abundant and deficient numbers ...........

A numbernis abundant, perfect, or deficient if the sum of its proper
divisors (including 1 but excludingnitself) is greater than, equal to, or
less thann. If we denote byσ(n)the sum ofalldivisors ofn, including
1 andnitself, thennis abundant, perfect, or deficient according asσ(n)
is greater than, equal to, or less than 2 n. The advantage of usingσ(n)is
that it can be easily computed if we know hownfactors into primes:
Abundant numbers up to 200:

12 18 20 24 30 36 40 42 48 54 56 60 66 70 72

78 80 84 88 90 96 100 102 104 108 112 114 120 126 132

138 140 144 150 156 160 162 168 174 176 180 186 192 196 198

200

Deficientevennumbers up to 200:

2 4 8 10 14 16 22 26 32 34 38 44 46 50 52

58 62 64 68 74 76 82 86 92 94 98 106 110 116 118

122 124 128 130 134 136 142 146 148 152 154 158 164 166 170

172 178 182 184 188 190 194

All multiples of 6 are abundant. But not conversely. 20 is abundant.

945 is the first odd abundant number.

5775 and 5776 are the first pair of abundant numbers.

Pairs of consecutive abundant numbers up to 10,000:

5775 , 5776 5984 , 5985 7424 , 7425 11024 , 11025

21735 ,21736 21944,21945 26144,26145 27404, 27405

39375 ,39376 43064,43065 49664,49665 56924, 56925

58695 ,58696 61424,61425 69615,69616 70784, 70785

76544 ,76545 77175,77176 79695,79696 81080, 81081

81675 ,81676 82004,82005 84524,84525 84644, 84645

89775 ,89776 91664,91665 98175, 98176 ...

The first triple of abundant numbers^3

n factorization σ(n) σ(n)− 2 n

171078830 2· 5 · 13 · 23 · 29 ·1973 358162560 16004900

171078831 3^3 · 7 · 11 · 19 · 61 · 71 342835200 677538

171078832 2^4 ·^31 ·^344917 342158656 992

(^3) Discovered in 1975 by Laurent Hodges and Reid, [Pickover, p.364].