# 26

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## Number

$26$ (twenty-six) is:

$2 \times 13$

The $3$rd heptagonal pyramidal number after $1$, $8$:
$26 = 1 + 7 + 18 = \dfrac {3 \left({3 + 1}\right) \left({5 \times 3 - 2}\right)} 6$

The $4$th second pentagonal number after $2$, $7$, $15$:
$26 = \dfrac {4 \left({3 \times 4 + 1}\right)} 2$

The $8$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$:
$26 = \dfrac {4 \left({3 \times 4 + 1}\right)} 2$

The $10$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$:
$26 = 2 \times 13$

The $11$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$:
$26 = 8 + 18$

The smallest non-palindromic integer whose square is palindromic:
$26^2 = 676$

The $2$nd nontotient after $14$:
$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 26$
where $\phi \left({m}\right)$ denotes the Euler $\phi$ function

The $2$nd noncototient after $10$:
$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 26$
where $\phi \left({m}\right)$ denotes the Euler $\phi$ function

The $11$th even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $20$, $22$ which cannot be expressed as the sum of $2$ composite odd numbers.

The $17$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$, $19$, $20$, $21$, $24$, $25$ which cannot be expressed as the sum of distinct pentagonal numbers.

The $1$st non-palindromic square root of a palindromic square:
$26^2 = 676$

The $6$th integer after $0$, $1$, $8$, $17$, $18$ equal to the sum of the digits of its cube:
$26^3 = 17 \, 576$, while $1 + 7 + 5 + 7 + 6 = 26$

Cannot be represented by the sum of less than $6$ hexagonal numbers:
$26 = 6 + 6 + 6 + 6 + 1 + 1$

## Historical Note

The main cultural significance of the number $26$ is that it is the number of distinct letters in the English alphabet:

$\texttt {A B C D E F G H I J K L M N O P Q R S T U V W X Y Z}$
$\texttt {a b c d e f g h i j k l m n o p q r s t u v w x y z}$