# 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:
- $181^2 - 7 = 32 \, 768 = 2^{15}$

## Also see

*Previous ... Next*: Numbers such that Divisor Count divides Phi divides Divisor Sum*Previous ... Next*: Consecutive Integers with Same Euler Phi Value*Previous ... Next*: Ramanujan-Nagell Number

*Previous ... Next*: Magic Constant of Magic Square*Previous ... Next*: Pentatope Number*Previous ... Next*: Bell Number

*Previous ... Next*: Hexagonal Number

*Previous ... Next*: Integers such that Difference with Power of 2 is always Prime*Previous ... Next*: Second Pentagonal Number*Previous ... Next*: Sum of 4 Consecutive Binomial Coefficients forming Square

*Previous ... Next*: Sum of Sequence of Alternating Positive and Negative Factorials being Prime*Previous ... Next*: Triangular Number

*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares*Previous ... Next*: Numbers not Sum of Distinct Squares*Previous ... Next*: Generalized Pentagonal Number*Previous ... Next*: Zuckerman Number

*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Previous ... Next*: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares*Previous ... Next*: Lucky Number

*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1*Previous ... Next*: Semiprime Number*Previous ... Next*: Consecutive Integers whose Product is Primorial

*Next*: Triangular Number Pairs with Triangular Sum and Difference*Next*: Fermat Pseudoprime to Base 4*Next*: Factorisation of Quintic x^5 - x + n into Irreducible Quadratic and Irreducible Cubic

## Historical Note

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

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $15$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $50,625$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $15$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $50,625$

Categories:

- Powers of 15/Examples
- Ramanujan-Nagell Numbers/Examples
- Count of All Permutations on n Objects/Examples
- Pentatope Numbers/Examples
- Bell Numbers/Examples
- Hexagonal Numbers/Examples
- Second Pentagonal Numbers/Examples
- Triangular Numbers/Examples
- Generalized Pentagonal Numbers/Examples
- Zuckerman Numbers/Examples
- Lucky Numbers/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Semiprimes/Examples
- Specific Numbers
- 15