# 116

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## Number

$116$ (one hundred and sixteen) is:

$2^2 \times 29$

The $9$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$:
$\nexists m \in \Z_{>0}: m - \map \phi m = 116$
where $\map \phi m$ denotes the Euler $\phi$ function

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

The $1$st of the $4$th pair of consecutive integers which both have $6$ divisors:
$\map \tau {116} = \map \tau {117} = 6$

The $22$nd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.

The $54$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $95$, $96$, $100$, $101$, $102$, $107$, $112$ which cannot be expressed as the sum of distinct pentagonal numbers.

The $11$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$