30

Number
$30$ (thirty) is:


 * $2 \times 3 \times 5$


 * The $1$st sphenic number:
 * $30 = 2 \times 3 \times 5$


 * The $4$th square pyramidal number after $1$, $5$, $14$:
 * $30 = 1 + 4 + 9 + 14 = \dfrac {4 \left({4 + 1}\right) \left({2 \times 4 + 1}\right)} 6$


 * The $5$th abundant number after $12$, $18$, $20$, $24$:
 * $1 + 2 + 3 + 5 + 6 + 10 + 15 = 42 > 30$


 * The $13$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$:
 * $\sigma \left({30}\right) = 72$


 * The $3$rd primorial after $1$, $2$, $6$ (counting $1$ as the zeroth):
 * $30 = p_3 \# = 5\# = 2 \times 3 \times 5$
 * Hence the smallest positive integer with $3$ distinct prime factors


 * The index (after $2$, $3$, $6$) of the $4$th Woodall prime:
 * $30 \times 2^{30} - 1$


 * The $7$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$:
 * $30 = 2 + 3 + 10 + 15$


 * The smallest Giuga number:
 * $\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 {30} = 1$


 * The $1$st of the smallest triplet of consecutive positive even integers $n$ with the property $n + \tau \left({n}\right) = m$ for some $m$:
 * $30 + \tau \left({30}\right) = 32 + \tau \left({32}\right) = 34 + \tau \left({34}\right) = 38$


 * The $6$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$ such that no factorial of an integer can end with $n$ zeroes


 * The $19$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$, $19$, $20$, $21$, $24$, $25$, $26$, $29$ which cannot be expressed as the sum of distinct pentagonal numbers.


 * The $4$th integer $n$ after $1$, $3$, $15$ with the property that $\tau \left({n}\right) \mathrel \backslash \phi \left({n}\right) \mathrel \backslash \sigma \left({n}\right)$:
 * $\tau \left({30}\right) = 8$, $\phi \left({30}\right) = 8$, $\sigma \left({30}\right) = 72$


 * The $3$rd primorial which can be expressed as the product of consecutive integers:
 * $30 = 5 \# = 5 \times 6$


 * The $7$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
 * $3$, $4$, $6$, $7$, $12$, $14$, $30$


 * The $17$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$, $20$, $21$, $24$, $27$:
 * $30 = 10 \times 3 = 10 \times \left({3 + 0}\right)$

Also see

 * Pythagorean Triangles whose Area equal their Perimeter
 * Smallest Consecutive Even Numbers such that Added to Divisor Count are Equal