# 32

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## Number

$32$ (thirty-two) is:

$2^5$

In binary:
$100 \, 000$

The $2$nd fifth power after $1$:
$32 = 2 \times 2 \times 2 \times 2 \times 2$

The $2$nd element of the $2$nd pair of integers $m$ whose values of $m \map \tau m$ is equal:
$24 \times \map \tau {24} = 192 = 32 \times \map \tau {32}$

The $5$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$:
$32 = 2^5$

The $6$th almost perfect number after $1$, $2$, $4$, $8$, $16$:
$\map \sigma {32} = 63 = 2 \times 32 - 1$

The $8$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$

The $8$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$

The $9$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$:
$32 \to 3^2 + 2^2 = 9 + 4 = 13 \to 1^2 + 3^2 = 9 + 1 = 10 \to 1^2 + 0^2 = 1$

The $13$th even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $20$, $22$, $26$, $28$ which cannot be expressed as the sum of $2$ composite odd numbers.

The $17$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $\ldots$

The $18$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

The $21$st positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$ which cannot be expressed as the sum of distinct pentagonal numbers.

The $22$nd integer $n$ such that $2^n$ contains no zero in its decimal representation:
$2^{32} = 4 \, 294 \, 967 \, 296$

## Historical Note

$32 \fahr$ is the melting point of water on the Fahrenheit scale.