# 22

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## Number

$22$ (twenty-two) is:

$2 \times 11$

The $1$st positive integer which can be expressed as the sum of $2$ odd primes in $3$ ways:
$22 = 19 + 3 = 17 + 5 = 11 + 11$

The $2$nd integer after $1$ which equals the number of digits in its factorial:
$22! = 1 \, 124 \, 000 \, 727 \, 777 \, 607 \, 680 \, 000$
which has $22$ digits

The $2$nd Smith number after $4$:
$2 + 2 = 2 + 1 + 1 = 4$

The $3$rd pentagonal number after $1$, $5$ which is also palindromic:
$22 = 1 + 4 + 7 + 10 = \dfrac {4 \left({3 \times 4 - 1}\right)} 2$

The $3$rd hexagonal pyramidal number after $1$, $7$:
$22 = 1 + 6 + 15$

The $3$rd number after $1$, $3$ whose $\sigma$ value is square:
$\sigma \left({22}\right) = 36 = 6^2$

The $4$th pentagonal number after $1$, $5$, $12$:
$22 = 1 + 4 + 7 + 10 = \dfrac {4 \left({3 \times 4 - 1}\right)} 2$

The $7$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$:
$22 = \dfrac {4 \left({3 \times 4 - 1}\right)} 2$

The $8$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$:
$22 = 2 \times 11$

The $8$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $\ldots$

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

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

$22$ is a palindromic number whose square is also palindromic:
$22^2 = 484$