15

Number
$15$ (fifteen) is:


 * The first product of two different odd primes:
 * $3 \times 5$


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


 * The $5$th triangular number after $1$, $3$, $6$, $10$:
 * $15 = 1 + 2 + 3 + 4 + 5 = \dfrac {5 \left({5 + 1}\right)} 2$


 * The $3$rd hexagonal number after $1$, $6$:
 * $15 = 1 + 5 + 9 = 3 \left({2 \times 3 - 1}\right)$


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


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


 * The $3$rd pentatope number after $1$, $5$:
 * $15 = 1 + 4 + 10 = \dfrac {3 \left({3 + 1}\right) \left({3 + 2}\right) \left({3 + 3}\right)} {24}$


 * 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 magic constant of the order $3$ magic square.
 * $15 = \dfrac {3 \left({3^2 + 1}\right)} 2$


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


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


 * 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 $13$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 $3$rd positive integer solution after $1$, $3$ to $\phi \left({n}\right) = \phi \left({n + 1}\right)$:
 * $\phi \left({15}\right) = 8 = \phi \left({16}\right)$


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


 * 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 $10$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 $3$rd integer $n$ after $1$, $3$ with the property that $\tau \left({n}\right) \mathrel \backslash \phi \left({n}\right) \mathrel \backslash \sigma \left({n}\right)$:
 * $\tau \left({15}\right) = 4$, $\phi \left({15}\right) = 8$, $\sigma \left({15}\right) = 24$


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


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


 * 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 first 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 $4$th Ramanujan-Nagell number after $0$, $1$, $3$:
 * $15 = 2^4 - 1 = \dfrac {5 \left({5 + 1}\right)} 2$


 * 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 an irreducible cubic:

Also see

 * Solutions of Ramanujan-Nagell Equation