42

Number
$42$ (forty-two) is:


 * $2 \times 3 \times 7$


 * The $2$nd sphenic number after $30$:
 * $42 = 2 \times 3 \times 7$


 * The magic constant of a magic cube of order $3$, after $1$, $(9)$:
 * $42 = \displaystyle \dfrac 1 {3^2} \sum_{k \mathop = 1}^{3^3} k = \dfrac {3 \paren {3^3 + 1} } 2$


 * The $4$th and largest positive integer after $1$, $3$, $14$ of which the product of its Euler $\phi$ function and its $\tau$ function equals its $\sigma$ function:
 * $\phi \left({42}\right) \tau \left({42}\right) = 12 \times 8 = 96 = \sigma \left({42}\right)$


 * The $5$th Catalan number after $(1)$, $1$, $2$, $5$, $14$:
 * $\dfrac 1 {5 + 1} \dbinom {2 \times 5} 5 = \dfrac 1 6 \times 252 = 42$


 * The $8$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$ such that no factorial of an integer can end with $n$ zeroes.


 * The $8$th abundant number after $12$, $18$, $20$, $24$, $30$, $36$, $40$:
 * $1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 > 42$


 * The $10$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$:
 * $42 = 7 + 14 + 21$


 * The $15$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$:
 * $\sigma \left({42}\right) = 96$


 * The $25$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$, $33$, $37$, $38$ which cannot be expressed as the sum of distinct pentagonal numbers.

Also see