10

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


 * The base of the decimal system


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


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


 * The $1$st power of $10$ after the zeroth $1$:
 * $10 = 10^1$


 * The $1$st noncototient:
 * $\nexists m \in \Z_{>0}: m - \map \phi m = 10$
 * where $\map \phi m$ denotes the Euler $\phi$ function


 * The $1$st positive integer with multiplicative persistence of $1$


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


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


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


 * The $2$nd after $1$ of the $5$ tetrahedral numbers which are also triangular


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


 * The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $2$ different ways:
 * $10 = 1 + 9 = 3 + 7$


 * 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 $3$rd tetrahedral number after $1$, $4$:
 * $10 = 1 + 3 + 6 = \dfrac {3 \paren {3 + 1} \paren {3 + 2} } 6$


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


 * The $3$rd after $2$, $5$ of $4$ numbers whose letters, when spelt in French, are in alphabetical order:
 * dix


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


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


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


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


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


 * The $5$th after $1$, $2$, $5$, $6$ of $6$ integers $n$ such that the alternating group $A_n$ is ambivalent


 * The $7$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$:
 * $\map {\sigma_1} {10} = 18$


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


 * The $7$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime:
 * $10! - 9! + 8! - 7! + 6! - 5! + 4! - 3! + 2! - 1! = 3 \, 301 \, 819$


 * 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 $9$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways


 * The $10$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$:
 * $10 = 10 \times 1 = 10 \times \paren {1 + 0}$


 * The $10$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$ such that $5^n$ contains no zero in its decimal representation:
 * $5^{10} = 9 \, 765 \, 625$


 * The total of all the entries in a magic square of order $2$ (if it were to exist), after $1$:
 * $10 = \ds \sum_{k \mathop = 1}^{2^2} k = \dfrac {2^2 \paren {2^2 + 1} } 2$

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

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