# 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 $1$, $(9)$:
$42 = \ds \dfrac 1 {3^2} \sum_{k \mathop = 1}^{3^3} k = \dfrac {3 \paren {3^3 + 1} } 2$

The $4$th and largest positive integer after $1$, $3$, $14$ of which the product of its Euler $\phi$ function and its divisor count equals its divisor sum:
$\map \phi {42} \map {\sigma_0} {42} = 12 \times 8 = 96 = \map {\sigma_1} {42}$

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

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 number of integer partitions for $10$:
$\map p {10} = 42$

The $15$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$:
$\map {\sigma_1} {42} = 96$

The $17$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$, $18$, $24$, $30$, $36$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

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.

The $29$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $27$, $30$, $31$, $32$, $35$, $36$, $37$, $38$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

### Arithmetic Functions on $42$

 $\ds \map {\sigma_0} { 42 }$ $=$ $\ds 8$ $\sigma_0$ of $42$ $\ds \map \phi { 42 }$ $=$ $\ds 12$ $\phi$ of $42$ $\ds \map {\sigma_1} { 42 }$ $=$ $\ds 96$ $\sigma_1$ of $42$

## 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.