105

Number
$105$ (one hundred and five) is:


 * $3 \times 5 \times 7$


 * The $14$th triangular number after $1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91$:
 * $105 = \displaystyle \sum_{k \mathop = 1}^{14} k = \dfrac {14 \times \left({14 + 1}\right)} 2$


 * The $23$rd lucky number:
 * $1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 73, 75, 79, 87, 93, 99, 105, \ldots$


 * The $1$st of the $1$st ordered triple of consecutive integers that have Euler $\phi$ values which are strictly increasing:
 * $\phi \left({105}\right) = 48$, $\phi \left({106}\right) = 52$, $\phi \left({107}\right) = 106$


 * The $2$nd of the $2$nd ordered quadruple of consecutive integers that have sigma values which are strictly decreasing:
 * $\sigma \left({104}\right) = 210, \ \sigma \left({105}\right) = 192, \ \sigma \left({106}\right) = 162, \ \sigma \left({107}\right) = 108$


 * The $7$th positive integer $n$ after $4, 7, 15, 21, 45, 75$, and largest known, such that $n - 2^k$ is prime for all $k$


 * The largest integer such that all smaller odd integers greater than $1$ which are coprime to it are prime


 * The smallest positive integer $n$ such that $1$ can be expressed as the sum of reciprocals of distinct odd integers such that none are less than $\dfrac 1 n$:
 * $1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {33} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {55} + \dfrac 1 {77} + \dfrac 1 {105}$


 * The $2$nd positive integer $n$ such that $\sigma \left({n}\right) = \dfrac {\phi \left({n}\right) \times \tau \left({n}\right)} 2$:
 * $\sigma \left({105}\right) = 192 = \dfrac {\phi \left({105}\right) \times \tau \left({105}\right)} 2$


 * The $9$th integer $n$ after $1, 3, 15, 30, 35, 56, 70, 78$ with the property that $\tau \left({n}\right) \mathrel \backslash \phi \left({n}\right) \mathrel \backslash \sigma \left({n}\right)$:
 * $\tau \left({105}\right) = 8$, $\phi \left({105}\right) = 48$, $\sigma \left({105}\right) = 192$

Also see

 * Largest Integer whose Smaller Odd Coprimes are Prime
 * Reciprocals of Odd Numbers adding to 1