# 15

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## Number

$15$ (fifteen) is:

The $1$st product of two different odd primes:
$3 \times 5$

The $1$st power of $15$ after the zeroth $1$:
$15 = 15^1$

The $1$st Fermat pseudoprime to base $4$:
$4^{15} \equiv 4 \pmod {15}$

The $1$st positive integer whose $4$th power can be expressed as the sum of $5$ distinct $4$th powers:
$15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$

The $1$st element of the $1$st pair of triangular numbers whose sum and difference are also both triangular:
$15 = T_5$, $21 = T_6$, $15 + 21 = T_8$, $21 - 15 = T_3$

The $1$st positive integer $n$ such that $\pm n$ allows $x^5 - x + n$ to be factorized into the product of an irreducible quadratic and an irreducible cubic:
$x^5 - x + 15 = \paren {x^2 + x + 3} \paren {x^3 - x^2 - 2x + 5}$
$x^5 - x - 15 = \paren {x^2 - x + 3} \paren {x^3 + x^2 - 2x - 5}$

The $3$rd hexagonal number after $1$, $6$:
$15 = 1 + 5 + 9 = 3 \paren {2 \times 3 - 1}$

The $3$rd second pentagonal number after $2$, $7$:
$15 = \dfrac {3 \paren {3 \times 3 + 1} } 2$

The $3$rd pentatope number after $1$, $5$:
$15 = 1 + 4 + 10 = \dfrac {3 \paren {3 + 1} \paren {3 + 2} \paren {3 + 3} } {24}$

The $3$rd positive integer solution after $1$, $3$ to $\map \phi n = \map \phi {n + 1}$:
$\map \phi {15} = 8 = \map \phi {16}$

The $3$rd positive integer $n$ after $4$, $7$ such that $n - 2^k$ is prime for all $k$

The $3$rd integer $n$ after $1$, $3$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
$\map {\sigma_0} {15} = 4$, $\map \phi {15} = 8$, $\map {\sigma_1} {15} = 24$

The magic constant of a magic square of order $3$, after $1$, $(5)$:
$15 = \ds \dfrac 1 3 \sum_{k \mathop = 1}^{3^2} k = \dfrac {3 \paren {3^2 + 1} } 2$

The total number of permutations of $r$ objects from a set of $3$ objects, where $1 \le r \le 3$

The $2$nd of the $4$th pair of consecutive integers whose product is a primorial:
$14 \times 15 = 210 = 7 \#$

The $4$th Ramanujan-Nagell number after $0$, $1$, $3$:
$15 = 2^4 - 1 = \dfrac {5 \paren {5 + 1} } 2$

The $4$th Bell number after $(1)$, $1$, $2$, $5$

The $5$th triangular number after $1$, $3$, $6$, $10$:
$15 = 1 + 2 + 3 + 4 + 5 = \dfrac {5 \paren {5 + 1} } 2$

The $5$th integer $n$ after $-1$, $0$, $2$, $7$ such that $\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$ for integer $m$:
$\dbinom {15} 0 + \dbinom {15} 1 + \dbinom {15} 2 + \dbinom {15} 3 = 24^2$

The $6$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$:
$15 = \dfrac {3 \paren {3 \times 3 + 1} } 2$

The $6$th semiprime after $4$, $6$, $9$, $10$, $14$:
$15 = 3 \times 5$

The $6$th lucky number:
$1$, $3$, $7$, $9$, $13$, $15$, $\ldots$

The $6$th odd positive integer after $1$, $3$, $5$, $7$, $9$ such that all smaller odd integers greater than $1$ which are coprime to it are prime

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

The $8$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$ 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:
$15! - 14! + 13! - 12! + 11! - 10! + 9! - 8! + 7! - 6! + 5! - 4! + 3! - 2! + 1! = 1 \, 226 \, 280 \, 710 \, 981$

The $8$th odd positive integer after $1$, $3$, $5$, $7$, $9$, $11$, $13$ that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime

The $10$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$, $10$, $12$ which cannot be expressed as the sum of exactly $5$ non-zero squares.

The $10$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

The $10$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$ which cannot be expressed as the sum of distinct pentagonal numbers.

The $12$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$:
$15 = 3 \times 5 = 3 \times \paren {1 \times 5}$

The $13$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$, $12$, $13$, $14$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

The $13$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$ such that $2^n$ contains no zero in its decimal representation:
$2^{15} = 32 \, 768$

The largest $n$ for which the Ramanujan-Nagell equation $x^2 - 7 = 2^n$ has an integral solution:
$181^2 - 7 = 32 \, 768 = 2^{15}$

## Historical Note

There are $15$ red balls in a game of snooker, arranged in an equilateral triangle.