15

Number
$15$ (fifteen) is:


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


 * 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 $10$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:
 * $14 \times 15 = 210 = 7 \#$

Also see

 * Solutions of Ramanujan-Nagell Equation