6

Number
$6$ (six) is:


 * The $2$nd composite number, and the first with distinct prime factors:
 * $6 = 2 \times 3$


 * Hence the first positive integer after $1$ which is not the power of a prime number.


 * 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 numbers:
 * $6 = 1 + 2 + 3 = 1 \times 2 \times 3$


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


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


 * The $4$th highly composite number after $1, 2, 4$:
 * $\tau \left({6}\right) = 4$


 * The $4$th superabundant number after $1, 2, 4$:
 * $\dfrac {\sigma \left({6}\right)} 6 = 2$


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


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


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


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


 * The $2$nd hexagonal number after $1$:
 * $6 = 1 + 5 = 2 \left({2 \times 2 - 1}\right)$


 * The $3$rd triangular number after $1, 3$:
 * $6 = 1 + 2 + 3 = \dfrac {3 \left({3 + 1}\right)} 2$


 * The $2$nd pentagonal pyramidal number after $1$:
 * $6 = 1 + 5 = \dfrac {2^2 \left({2 + 1}\right)} 2$


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


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


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


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


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


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


 * The $3$rd palindromic triangular number after $1, 3$.


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


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


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

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