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 smallest 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:
 * $6 = 1 + 2 + 3$

$2$nd Term

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


 * 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 $1$, $2$, $4$:
 * $\map \tau 6 = 4$


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


 * 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 $m = \displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime:
 * $6! - 5! + 4! - 3! + 2! - 1! = 619$

$5$th Term

 * The $5$th highly abundant number after $1$, $2$, $3$, $4$:
 * $\map \sigma 6 = 12$


 * 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

$6$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 $1$, $2$, $3$, $4$, $5$:
 * $6 = 1 \times 6$

$7$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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