105

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


 * $3 \times 5 \times 7$


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


 * The $2$nd positive integer $n$ such that $\map {\sigma_1} n = \dfrac {\map \phi n \times \map {\sigma_0} n} 2$:
 * $\map {\sigma_1} {105} = 192 = \dfrac {\map \phi {105} \times \map {\sigma_0} {105} } 2$


 * The $1$st of the $2$nd pair of triangular numbers whose sum and difference are also both triangular:
 * $105 = T_{14}$, $171 = T_{18}$, $105 + 171 = T_{23}$, $171 - 105 = T_{11}$


 * The $2$nd of the $2$nd ordered quadruple of consecutive integers that have divisor sums which are strictly decreasing:
 * $\map {\sigma_1} {104} = 210, \ \map {\sigma_1} {105} = 192, \ \map {\sigma_1} {106} = 162, \ \map {\sigma_1} {107} = 108$


 * The $4$th integer after $1$, $14$, $30$ whose divisor sum divided by its Euler $\phi$ value is a square:
 * $\dfrac {\map {\sigma_1} {105} } {\map \phi {105} } = \dfrac {192} {48} = 4 = 2^2$


 * 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 $7$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$:
 * $105 = 3 \times 5 \times 7$


 * The $9$th integer $n$ after $1$, $3$, $15$, $30$, $35$, $56$, $70$, $78$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
 * $\map {\sigma_0} {105} = 8$, $\map \phi {105} = 48$, $\map {\sigma_1} {105} = 192$


 * The $9$th and largest odd positive integer after $1$, $3$, $5$, $7$, $9$, $15$, $21$, $45$ such that all smaller odd integers greater than $1$ which are coprime to it are prime.


 * The $13$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime


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


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

Also see

 * Reciprocals of Odd Numbers adding to 1