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:
- $6^2 = 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = \dfrac {8 \paren {8 + 1} } 2$
$1$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:
- 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 $1$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 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:
- $6 = T_3$
$5$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 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 $6$
\(\displaystyle \map \tau { 6 }\) | \(=\) | \(\displaystyle 4\) | $\tau$ of $6$ | ||||||||||
\(\displaystyle \map \phi { 6 }\) | \(=\) | \(\displaystyle 2\) | $\phi$ of $6$ | ||||||||||
\(\displaystyle \map \sigma { 6 }\) | \(=\) | \(\displaystyle 12\) | $\sigma$ of $6$ |
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
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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:
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
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6$
Categories:
- Work To Do
- Hexagonal Numbers/Examples
- Pyramidal Numbers/Examples
- Ore Numbers/Examples
- Powers of 6/Examples
- Special Highly Composite Numbers/Examples
- Central Binomial Coefficients/Examples
- Factorials/Examples
- Primorials/Examples
- Triangular Numbers/Examples
- Woodall Primes/Examples
- Ulam Numbers/Examples
- Highly Abundant Numbers/Examples
- Semiprimes/Examples
- Highly Composite Numbers/Examples
- Superabundant Numbers/Examples
- Pluperfect Digital Invariants/Examples
- Zuckerman Numbers/Examples
- Harshad Numbers/Examples
- Trimorphic Numbers/Examples
- Automorphic Numbers/Examples
- Primitive Semiperfect Numbers/Examples
- Semiperfect Numbers/Examples
- Perfect Numbers/Examples
- Unitary Perfect Numbers/Examples
- Specific Numbers
- 6