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 $\tau \left({n}\right) \mathrel \backslash \phi \left({n}\right) \mathrel \backslash \sigma \left({n}\right)$:
 * $\tau \left({105}\right) = 8$, $\phi \left({105}\right) = 48$, $\sigma \left({105}\right) = 192$


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

Also see

 * Tau Function of 105: $\tau \left({105}\right) = 8$
 * Euler Phi Function of 105: $\phi \left({105}\right) = 48$
 * Sigma of 105: $\sigma \left({105}\right) = 192$


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