# 6

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## Number

$6$ (six) is:

$2 \times 3$

The only triangular number with less than $660$ digits whose square is also triangular:
$6^2 = 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = \dfrac {8 \paren {8 + 1} } 2$

The only positive integer which is the sum of exactly $3$ of its distinct coprime divisors

### $1$st Term

The $1$st positive integer which can be expressed as the sum of $2$ odd primes in $1$ way:
$6 = 3 + 3$

The $1$st triangular number which can be expressed as the product of $3$ consecutive integers:
$6 = T_3 = 1 \times 2 \times 3$

The $1$st power of $6$ after the zeroth $1$:
$6 = 6^1$

The $1$st:
perfect number
semiperfect number
primitive semiperfect number
and the only number which is the sum and product of the same $3$ distinct positive integers:
$6 = 1 + 2 + 3 = 1 \times 2 \times 3$

The $1$st unitary perfect number:
$6 = 1 + 2 + 3$

The $1$st positive integer whose cube can be expressed as the sum of $3$ positive cube numbers:
$6^3 = 3^3 + 4^3 + 5^3$

### $2$nd Term

The $2$nd Ore number after $1$:
$\dfrac {6 \times \map {\sigma_0} 6} {\map {\sigma_1} 6} = 2$
and the $2$nd after $1$ whose divisors also have an arithmetic mean which is an integer:
$\dfrac {\map {\sigma_1} 6} {\map {\sigma_0} 6} = 3$

The $2$nd semiprime after $4$:
$6 = 2 \times 3$

The $2$nd primorial after $1$, $2$ (counting $1$ as the zeroth):
$6 = p_2 \# = 3 \# = 2 \times 3$

The $2$nd hexagonal number after $1$:
$6 = 1 + 5 = 2 \paren {2 \times 2 - 1}$

The $2$nd pentagonal pyramidal number after $1$:
$6 = 1 + 5 = \dfrac {2^2 \paren {2 + 1} } 2$

The $2$nd composite number, and the first with distinct prime factors:
$6 = 2 \times 3$
Hence the $1$st positive integer after $1$ which is not the power of a prime number.

The $2$nd primorial which can be expressed as the product of consecutive integers:
$3 \# = 6 = 2 \times 3$

The $2$nd central binomial coefficient after $2$:
$6 = \dbinom {2 \times 2} 2 := \dfrac {4!} {\paren {2!}^2}$

The $2$nd of the $3$rd pair of consecutive integers whose product is a primorial:
$5 \times 6 = 30 = 5 \#$

### $3$rd Term

The $3$rd special highly composite number after $1$, $2$

The $3$rd factorial after $1$, $2$:
$6 = 3! = 3 \times 2 \times 1$

The $3$rd triangular number after $1$, $3$:
$6 = 1 + 2 + 3 = \dfrac {3 \paren {3 + 1} } 2$

The $3$rd automorphic number after $1$, $5$:
$6^2 = 3 \mathbf 6$

The $3$rd even number after $2$, $4$ which cannot be expressed as the sum of $2$ composite odd numbers

The index (after $2, 3$) of the $3$rd Woodall prime:
$6 \times 2^6 - 1 = 383$

The $3$rd positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $\ldots$

The $3$rd integer $m$ after $3$, $4$ such that $m! - 1$ (its factorial minus $1$) is prime:
$6! - 1 = 720 - 1 = 719$

### $4$th Term

The $4$th highly composite number after $1$, $2$, $4$:
$\map {\sigma_0} 6 = 4$

The $4$th superabundant number after $1$, $2$, $4$:
$\dfrac {\map {\sigma_1} 6} 6 = \dfrac {12} 6 = 2$

The $4$th after $1$, $2$, $5$ of $6$ integers $n$ such that the alternating group $A_n$ is ambivalent

The $4$th trimorphic number after $1$, $4$, $5$:
$6^3 = 21 \mathbf 6$

The $4$th palindromic triangular number after $0$, $1$, $3$

The $4$th palindromic triangular number after $0$, $1$, $3$ whose index is itself palindromic:
$6 = T_3$

The $4$th integer $n$ after $3$, $4$, $5$ 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:
$6! - 5! + 4! - 3! + 2! - 1! = 619$

### $5$th Term

The $5$th highly abundant number after $1$, $2$, $3$, $4$:
$\map {\sigma_1} 6 = 12$

The $5$th Ulam number after $1$, $2$, $3$, $4$:
$6 = 2 + 4$

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

The $5$th positive integer after $1$, $2$, $3$, $4$ such that all smaller positive integers coprime to it are prime

The $5$th after $1$, $2$, $4$, $5$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes

The $5$th (strictly) positive integer after $1$, $2$, $3$, $4$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

### $6$th Term

The $6$th of the (trivial $1$-digit) pluperfect digital invariants after $1$, $2$, $3$, $4$, $5$:
$6^1 = 6$

The $6$th of the (trivial $1$-digit) Zuckerman numbers after $1$, $2$, $3$, $4$, $5$:
$6 = 1 \times 6$

The $6$th of the (trivial $1$-digit) harshad numbers after $1$, $2$, $3$, $4$, $5$:
$6 = 1 \times 6$

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

### $7$th Term

The $7$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$ such that $2^n$ contains no zero in its decimal representation:
$2^6 = 64$

The $7$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$ such that $5^n$ contains no zero in its decimal representation:
$5^6 = 15 \, 625$

The $7$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$ such that both $2^n$ and $5^n$ have no zeroes:
$2^6 = 64$, $5^6 = 15 \, 625$

The $7$th integer after $0$, $1$, $2$, $3$, $4$, $5$ which is (trivially) the sum of the increasing powers of its digits taken in order:
$6^1 = 6$

### Miscellaneous

$6 = \sqrt {1^3 + 2^3 + 3^3}$

The number of faces of a cube

The number of vertices of its dual, the regular octahedron

The number of edges of a tetrahedron

The area and semiperimeter of the $3-4-5$ triangle:
$6 = \dfrac {3 \times 4} 2 = \dfrac {3 + 4 + 5} 2$

### Arithmetic Functions on $6$

 $\ds \map {\sigma_0} { 6 }$ $=$ $\ds 4$ $\sigma_0$ of $6$ $\ds \map \phi { 6 }$ $=$ $\ds 2$ $\phi$ of $6$ $\ds \map {\sigma_1} { 6 }$ $=$ $\ds 12$ $\sigma_1$ of $6$

## Historical Note

The number $6$ was associated by the Pythagoreans with health.

It was also, like $5$, associated with marriage, as it is the product of $2$, the first female number, and $3$, the first male number.

It also stood for equilibrium, its symbol being two triangles placed base to base:

The number $6$ was also believed by the Pythagoreans to be perfect, as it was the sum of:

$1$, the unity
$2$, the principle of diversity
$3$, the "sacred trinity".

The fact that $1 + 2 + 3 = 1 \times 2 \times 3$ added to its mystique.

Six is a perfect number in itself ... God created all things in six days because this number is perfect. And it would remain perfect, even if the work of six days did not exist.
-- as reported by Ludwig Bieler

## Linguistic Note

Words derived from or associated with the number $6$ include:

sextet: a musical piece for, or a group of, $6$ musicians