6

== 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:
::$6 = \dfrac {3 \times 4} 2 = \dfrac {3 + 4 + 5} 2$
== Also see ==
* Perfect Number is Sum of Successive Odd Cubes except 6
* Prime equals Plus or Minus One modulo 6
* Only Number which is Sum of 3 Factors is 6
* Triangular Number whose Square is Triangular
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=== Next in Sequence: $9$ and above ===
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== Historical Note ==
== Linguistic Note ==
== Sources ==
* : $5$
* : $6$
* : $5$
* : $6$
Category:Specific Numbers
Category:6