# 30

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## Number

$30$ (thirty) is:

$2 \times 3 \times 5$

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

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

The $1$st element of the $1$st triplet of consecutive positive even integers $n$ with the property $n + \map \tau n = m$ for some $m$:
$30 + \map \tau {30} = 32 + \map \tau {32} = 34 + \map \tau {34} = 38$

The area and perimeter of the $2$nd of the only $2$ Pythagorean triples which define a Pythagorean triangle whose area equals its perimeter:
$\tuple {5, 12, 13}$

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

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

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

The $4$th integer $n$ after $1$, $3$, $15$ with the property that $\map \tau n \divides \map \phi n \divides \map \sigma n$:
$\map \tau {30} = 8$, $\map \phi {30} = 8$, $\map \sigma {30} = 72$

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

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

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

The $10$th and largest positive integer after $1$, $2$, $3$, $4$, $6$, $8$, $12$, $18$, $24$ such that all smaller positive integers coprime to it are prime

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

The $13$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $\ldots$

The $14$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$, $11$, $17$, $18$ such that $5^n$ contains no zero in its decimal representation:
$5^{30} = 931 \, 322 \, 574 \, 615 \, 478 \, 515 \, 625$

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

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.

### Arithmetic Functions on $30$

 $\displaystyle \map \tau { 30 }$ $=$ $\displaystyle 8$ $\tau$ of $30$ $\displaystyle \map \phi { 30 }$ $=$ $\displaystyle 8$ $\phi$ of $30$ $\displaystyle \map \sigma { 30 }$ $=$ $\displaystyle 72$ $\sigma$ of $30$