Divisor Counting Function/Examples

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Examples of Divisor Counting ($\tau$) Function

$\tau$ of $1$

The value of the $\tau$ function for the integer $1$ is $1$.

$\tau$ of $3$

$\map \tau 3 = 2$

$\tau$ of $12$

$\tau \left({12}\right) = 6$

$\tau$ of $60$

$\tau \left({60}\right) = 12$

$\tau$ of $105$

$\map \tau {105} = 8$

$\tau$ of $108$

$\tau \left({108}\right) = 12$

$\tau$ of $110$

$\tau \left({110}\right) = 8$

$\tau$ of $120$

$\tau \left({120}\right) = 16$

Table of Values of Divisor Counting ($\tau$) Function

The divisor counting ($\tau$) function for the first $200$ positive integers is as follows:

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

$\begin{array} {|r|r|} \hline n & \map \tau n \\ \hline 101 & 2 \\ 102 & 8 \\ 103 & 2 \\ 104 & 8 \\ 105 & 8 \\ 106 & 4 \\ 107 & 2 \\ 108 & 12 \\ 109 & 2 \\ 110 & 8 \\ 111 & 4 \\ 112 & 10 \\ 113 & 2 \\ 114 & 8 \\ 115 & 4 \\ 116 & 6 \\ 117 & 6 \\ 118 & 4 \\ 119 & 4 \\ 120 & 16 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map \tau n \\ \hline 121 & 3 \\ 122 & 4 \\ 123 & 4 \\ 124 & 6 \\ 125 & 4 \\ 126 & 12 \\ 127 & 2 \\ 128 & 8 \\ 129 & 4 \\ 130 & 8 \\ 131 & 2 \\ 132 & 12 \\ 133 & 4 \\ 134 & 4 \\ 135 & 8 \\ 136 & 8 \\ 137 & 2 \\ 138 & 8 \\ 139 & 2 \\ 140 & 12 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map \tau n \\ \hline 141 & 4 \\ 142 & 4 \\ 143 & 4 \\ 144 & 15 \\ 145 & 4 \\ 146 & 4 \\ 147 & 6 \\ 148 & 6 \\ 149 & 2 \\ 150 & 12 \\ 151 & 2 \\ 152 & 8 \\ 153 & 6 \\ 154 & 8 \\ 155 & 4 \\ 156 & 12 \\ 157 & 2 \\ 158 & 4 \\ 159 & 4 \\ 160 & 12 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map \tau n \\ \hline 161 & 4 \\ 162 & 10 \\ 163 & 2 \\ 164 & 6 \\ 165 & 8 \\ 166 & 4 \\ 167 & 2 \\ 168 & 16 \\ 169 & 3 \\ 170 & 8 \\ 171 & 6 \\ 172 & 6 \\ 173 & 2 \\ 174 & 8 \\ 175 & 6 \\ 176 & 10 \\ 177 & 4 \\ 178 & 4 \\ 179 & 2 \\ 180 & 18 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map \tau n \\ \hline 181 & 2 \\ 182 & 8 \\ 183 & 4 \\ 184 & 8 \\ 185 & 4 \\ 186 & 8 \\ 187 & 4 \\ 188 & 6 \\ 189 & 8 \\ 190 & 8 \\ 191 & 2 \\ 192 & 14 \\ 193 & 2 \\ 194 & 4 \\ 195 & 8 \\ 196 & 9 \\ 197 & 2 \\ 198 & 12 \\ 199 & 2 \\ 200 & 12 \\ \hline \end{array}$