42

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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.


Historical Note

Famously, the answer to life, the universe and everything, in The Hitchhiker's Guide to the Galaxy by Douglas Adams, is $42$.

The number is completely arbitrary. Adams looked out of the window, thought of a number, and decided "$42$. That'll do." Nothing more profound than that.


Also see

No further terms of this sequence are documented on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Sources