# 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

*Previous ... Next*: Catalan Number*Previous*: Integers whose Phi times Divisor Count equal Divisor Sum

*Previous ... Next*: Integer Partition*Previous ... Next*: Sphenic Number

*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Numbers of Zeroes that Factorial does not end with*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum

*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $42$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $42$