30

Number
$30$ (thirty) is:


 * $2 \times 3 \times 5$


 * 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):
 * $30 = 5\# = 2 \times 3 \times 5$
 * 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 $\tau \left({n}\right) \mathrel \backslash \phi \left({n}\right) \mathrel \backslash \sigma \left({n}\right)$:
 * $\tau \left({30}\right) = 8$, $\phi \left({30}\right) = 8$, $\sigma \left({30}\right) = 72$

Also see

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