26
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Number
$26$ (twenty-six) is:
- $2 \times 13$
- The $1$st non-palindromic integer whose square is palindromic:
- $26^2 = 676$
- The $2$nd nontotient after $14$:
- $\nexists m \in \Z_{>0}: \map \phi m = 26$
- where $\map \phi m$ denotes the Euler $\phi$ function
- The $2$nd noncototient after $10$:
- $\nexists m \in \Z_{>0}: m - \map \phi m = 26$
- where $\map \phi m$ denotes the Euler $\phi$ function
- The $3$rd heptagonal pyramidal number after $1$, $8$:
- $26 = 1 + 7 + 18 = \dfrac {3 \paren {3 + 1} \paren {5 \times 3 - 2} } 6$
- The $3$rd positive integer after $1$, $24$ whose Euler $\phi$ value is equal to the product of its digits:
- $\map \phi {26} = 12 = 2 \times 6$
- The $4$th second pentagonal number after $2$, $7$, $15$:
- $26 = \dfrac {4 \paren {3 \times 4 + 1} } 2$
- The $6$th Dudeney number after $0$, $1$, $8$, $17$, $18$:
- $26^3 = 17 \, 576$, while $1 + 7 + 5 + 7 + 6 = 26$
- The $8$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$:
- $26 = \dfrac {4 \paren {3 \times 4 + 1} } 2$
- 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 $11$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 $20$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $20$, $21$, $24$, $25$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
- Cannot be represented by the sum of less than $6$ hexagonal numbers:
- $26 = 6 + 6 + 6 + 6 + 1 + 1$
Arithmetic Functions on $26$
\(\ds \map \phi { 26 }\) | \(=\) | \(\ds 12\) | $\phi$ of $26$ | |||||||||||
\(\ds \map {\sigma_1} { 26 }\) | \(=\) | \(\ds 42\) | $\sigma_1$ of $26$ |
Also see
- Previous ... Next: Noncototient
- Previous ... Next: Nontotient
- Previous ... Next: Dudeney Number
- Previous ... Next: Ulam Number
- Previous ... Next: Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers
- Previous ... Next: Generalized Pentagonal Number
- Previous ... Next: Integers not Expressible as Sum of Distinct Primes of form 6n-1
- Previous ... Next: Numbers not Expressible as Sum of Distinct Pentagonal Numbers
- Previous ... Next: Semiprime Number
Historical Note
The main cultural significance of the number $26$ is that it is the number of distinct letters in the English alphabet:
- $\texttt {A B C D E F G H I J K L M N O P Q R S T U V W X Y Z}$
- $\texttt {a b c d e f g h i j k l m n o p q r s t u v w x y z}$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $26$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $26$