# 21

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## Number

$21$ (twenty-one) is:

$3 \times 7$

The smallest integer with $2$ distinct prime factors neither of which is a divisor of $10$

The number of parts in the smallest perfect square dissection of an integer square

The $1$st of the $5$ $2$-digit positive integers which can occur as a $5$-fold repetition at the end of a square number, for example:
$508 \, 853 \, 989^2 = 258 \, 932 \, 382 \, 121 \, 212 \, 121$

The second of the $1$st pair of triangular numbers whose sum and difference are also both triangular:
$15 = T_5$, $21 = T_6$, $15 + 21 = T_8$, $21 - 15 = T_3$

The $3$rd octagonal number after $1$, $8$:
$21 = 1 + 7 + 13 = 3 \paren {3 \times 3 - 2}$

The smallest number which can be expressed as the sum of at most $3$ triangular numbers in $4$ ways:
$21 = 15 + 6 = 15 + 3 + 3 = 10 + 10 + 1$

The $4$th positive integer $n$ after $4$, $7$, $15$ such that $n - 2^k$ is prime for all $k$

The $4$th after $0$, $1$, $3$ of the $5$ Fibonacci numbers which are also triangular

The $6$th triangular number, after $1$, $3$, $6$, $10$, $15$:
$21 = 1 + 2 + 3 + 4 + 5 + 6 = \dfrac {6 \times \paren {6 + 1} } 2$
Thus $21$ is the number of pips on a die

The $7$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$:
$21 = 3 \times 7$

The $7$th lucky number:
$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$

The $7$th odd positive integer after $1$, $3$, $5$, $7$, $9$, $15$ such that all smaller odd integers greater than $1$ which are coprime to it are prime

The $8$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$:
$21 = 8 + 13$

The $11$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $\ldots$

The $14$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$, $20$:
$21 = 7 \times 3 = 7 \times \paren {2 + 1}$

The $14$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$, $19$, $20$ which cannot be expressed as the sum of distinct pentagonal numbers

## Historical Note

There were $21$ shillings in $1$ guinea Sterling in pre-decimal British coinage.