117

Number
$117$ (one hundred and seventeen) is:


 * $3^2 \times 13$


 * The $9$th pentagonal number after $1, 5, 12, 22, 35, 51, 70, 92$:
 * $117 = 1 + 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 = \dfrac {9 \left({3 \times 9 - 1}\right)} 2$


 * The $17$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$, $51$, $57$, $70$, $77$, $92$, $100$:
 * $117 = \dfrac {9 \left({3 \times 9 - 1}\right)} 2$


 * The length of the $2$nd longest edge of the smallest cuboid whose edges and the diagonals of whose faces are all integers:
 * The lengths of the edges are $44, 117, 240$
 * The lengths of the diagonals of the faces are $125, 244, 267$.


 * The $3$rd term of the $2$nd $5$-tuple of consecutive integers have the property that they are not values of the $\sigma$ function $\sigma \left({n}\right)$ for any $n$:
 * $\left({115, 116, 117, 118, 119}\right)$


 * The $2$nd of the $4$th pair of consecutive integers which both have $6$ divisors:
 * $\tau \left({116}\right) = \tau \left({117}\right) = 6$


 * The $5$th of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
 * $73, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270$

Also see

 * Cuboid with Integer Edges and Face Diagonals