# 117

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## 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 $72$:
$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$