10

Number
$10$ (ten) is:
 * $2 \times 5$


 * The $2$nd number after $5$ to be the sum of two different squares:
 * $10 = 1^2 + 3^2$


 * The $4$th semiprime after $4$, $6$, $9$:
 * $10 = 2 \times 5$


 * The $3$rd number after $2$ and $6$ that is not the difference of two squares, as it is of the form $4 n + 2$:
 * $10 = 4 \times 2 + 2$


 * The $4$th triangular number after $1$, $3$, $6$:
 * $10 = 1 + 2 + 3 + 4 = \dfrac {4 \left({4 + 1}\right)} 2$


 * The only triangular number which is the sum of consecutive odd squares:
 * $10 = 1^2 + 3^2$


 * The $3$rd tetrahedral number after $1$ and $4$:
 * $10 = 1 + 3 + 6 = \dfrac {3 \left({3 + 1}\right) \left({3 + 2}\right)} 6$


 * The $7$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$:
 * $\sigma \left({10}\right) = 18$


 * The $3$rd happy number after $1$, $7$:
 * $10 \to 1^2 + 0^2 = 1$


 * The $4$th term of Göbel's sequence after $1$, $2$, $3$, $5$:
 * $10 = \left({1 + 1^2 + 2^2 + 3^2 + 5^2}\right) / 4$


 * Probably the only number (except for the obvious $\left({n!}\right)! = n! \left({n! - 1}\right)!$ whose factorial is the product of $2$ factorials:
 * $10! = 7! \, 6!$
 * and so consequently:
 * $10! = 7! \, 5! \, 3!$


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


 * The second positive integer after $1$ which is not the sum of a square and a prime:
 * $10 = 1 + 9 = 4 + 6 = 9 + 1$: none of $1$, $6$ and $9$ are prime.


 * The smallest positive integer with multiplicative persistence of $1$.


 * The base of the decimal system.


 * The $8$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$ which cannot be expressed as the sum of exactly $5$ non-zero squares.


 * The $5$th even number after $2$, $4$, $6$, $8$ which cannot be expressed as the sum of $2$ composite odd numbers.


 * The $1$st positive integer which can be expressed as the sum of $2$ odd primes in $2$ ways:
 * $10 = 3 + 7 = 5 + 5$


 * The $2$nd positive integer which cannot be expressed as the sum of a square and a prime:
 * $1$, $10$, $\ldots$


 * The $1$st non-square positive integer which cannot be expressed as the sum of a square and a prime


 * The $7$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$ which cannot be expressed as the sum of distinct pentagonal numbers.


 * The smallest positive integer having a multiplicative persistence of $2$.


 * The $3$rd of $4$ numbers whose letters, when spelt in French, are in alphabetical order:
 * dix

Also see

 * Divisibility by 10
 * Divisibility by Power of 10


 * 10 is Only Triangular Number that is Sum of Consecutive Odd Squares


 * Factorial as Product of Two Factorials
 * Factorial as Product of Three Factorials