117

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


 * $3^2 \times 13$


 * 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 divisor sum function $\map {\sigma_1} n$ for any $n$:
 * $\tuple {115, 116, 117, 118, 119}$


 * The $2$nd of the $4$th pair of consecutive integers which both have $6$ divisors:
 * $\map {\sigma_0} {116} = \map {\sigma_0} {117} = 6$


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


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


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

Also see

 * Cuboid with Integer Edges and Face Diagonals