Definition:Divisor Sum Function

Definition

Let $n$ be an integer such that $n \ge 1$.

The divisor sum function $\map {\sigma_1} n$ is defined on $n$ as being the sum of all the positive integer divisors of $n$.

That is:

$\ds \map {\sigma_1} n = \sum_{d \mathop \divides n} d$

where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.

Sequence of Values of Divisor Sum Function

The divisor sum function for the first $200$ positive integers is as follows:

$\quad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 1 & 1 \\ 2 & 3 \\ 3 & 4 \\ 4 & 7 \\ 5 & 6 \\ 6 & 12 \\ 7 & 8 \\ 8 & 15 \\ 9 & 13 \\ 10 & 18 \\ 11 & 12 \\ 12 & 28 \\ 13 & 14 \\ 14 & 24 \\ 15 & 24 \\ 16 & 31 \\ 17 & 18 \\ 18 & 39 \\ 19 & 20 \\ 20 & 42 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 21 & 32 \\ 22 & 36 \\ 23 & 24 \\ 24 & 60 \\ 25 & 31 \\ 26 & 42 \\ 27 & 40 \\ 28 & 56 \\ 29 & 30 \\ 30 & 72 \\ 31 & 32 \\ 32 & 63 \\ 33 & 48 \\ 34 & 54 \\ 35 & 48 \\ 36 & 91 \\ 37 & 38 \\ 38 & 60 \\ 39 & 56 \\ 40 & 90 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 41 & 42 \\ 42 & 96 \\ 43 & 44 \\ 44 & 84 \\ 45 & 78 \\ 46 & 72 \\ 47 & 48 \\ 48 & 124 \\ 49 & 57 \\ 50 & 93 \\ 51 & 72 \\ 52 & 98 \\ 53 & 54 \\ 54 & 120 \\ 55 & 72 \\ 56 & 120 \\ 57 & 80 \\ 58 & 90 \\ 59 & 60 \\ 60 & 168 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 61 & 62 \\ 62 & 96 \\ 63 & 104 \\ 64 & 127 \\ 65 & 84 \\ 66 & 144 \\ 67 & 68 \\ 68 & 126 \\ 69 & 96 \\ 70 & 144 \\ 71 & 72 \\ 72 & 195 \\ 73 & 74 \\ 74 & 114 \\ 75 & 124 \\ 76 & 140 \\ 77 & 96 \\ 78 & 168 \\ 79 & 80 \\ 80 & 186 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 81 & 121 \\ 82 & 126 \\ 83 & 84 \\ 84 & 224 \\ 85 & 108 \\ 86 & 132 \\ 87 & 120 \\ 88 & 180 \\ 89 & 90 \\ 90 & 234 \\ 91 & 112 \\ 92 & 168 \\ 93 & 128 \\ 94 & 144 \\ 95 & 120 \\ 96 & 252 \\ 97 & 98 \\ 98 & 171 \\ 99 & 156 \\ 100 & 217 \\ \hline \end{array}$

$\quad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 101 & 102 \\ 102 & 216 \\ 103 & 104 \\ 104 & 210 \\ 105 & 192 \\ 106 & 162 \\ 107 & 108 \\ 108 & 280 \\ 109 & 110 \\ 110 & 216 \\ 111 & 152 \\ 112 & 248 \\ 113 & 114 \\ 114 & 240 \\ 115 & 144 \\ 116 & 210 \\ 117 & 182 \\ 118 & 180 \\ 119 & 144 \\ 120 & 360 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 121 & 133 \\ 122 & 186 \\ 123 & 168 \\ 124 & 224 \\ 125 & 156 \\ 126 & 312 \\ 127 & 128 \\ 128 & 255 \\ 129 & 176 \\ 130 & 252 \\ 131 & 132 \\ 132 & 336 \\ 133 & 160 \\ 134 & 204 \\ 135 & 240 \\ 136 & 270 \\ 137 & 138 \\ 138 & 288 \\ 139 & 140 \\ 140 & 336 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 141 & 192 \\ 142 & 216 \\ 143 & 168 \\ 144 & 403 \\ 145 & 180 \\ 146 & 222 \\ 147 & 228 \\ 148 & 266 \\ 149 & 150 \\ 150 & 372 \\ 151 & 152 \\ 152 & 300 \\ 153 & 234 \\ 154 & 288 \\ 155 & 192 \\ 156 & 392 \\ 157 & 158 \\ 158 & 240 \\ 159 & 216 \\ 160 & 378 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 161 & 192 \\ 162 & 363 \\ 163 & 164 \\ 164 & 294 \\ 165 & 288 \\ 166 & 252 \\ 167 & 168 \\ 168 & 480 \\ 169 & 183 \\ 170 & 324 \\ 171 & 260 \\ 172 & 308 \\ 173 & 174 \\ 174 & 360 \\ 175 & 248 \\ 176 & 372 \\ 177 & 240 \\ 178 & 270 \\ 179 & 180 \\ 180 & 546 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map {\sigma_1} n \\ \hline 181 & 182 \\ 182 & 336 \\ 183 & 248 \\ 184 & 360 \\ 185 & 228 \\ 186 & 384 \\ 187 & 216 \\ 188 & 336 \\ 189 & 320 \\ 190 & 360 \\ 191 & 192 \\ 192 & 508 \\ 193 & 194 \\ 194 & 294 \\ 195 & 336 \\ 196 & 399 \\ 197 & 198 \\ 198 & 468 \\ 199 & 200 \\ 200 & 465 \\ \hline \end{array}$

Also defined as

Some sources define the divisor sum function of $n$ as the aliquot sum of $n$, that is, sum of the proper divisors of $n$.

Also known as

The divisor sum function is usually referred to as the sigma function, and denoted $\map \sigma n$.

However, there is more than one function which bears that name, so on $\mathsf{Pr} \infty \mathsf{fWiki}$ we use the more explicit name divisor sum function.

Some sources refer to it as just the sum function, but that lacks precision.

Also see

• Definition:Divisor Function
• Definition:Divisor Count Function

• Results about the divisor sum function can be found here.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): Glossary
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): sigma function or $\sigma$ function: 1.
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): sigma function or $\sigma$ function: 2.
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): Glossary
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): sigma function (sum function)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): sigma function
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): sigma function