# 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 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:
- 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 $1$:
- $\dfrac {6 \times \map {\sigma_0} 6} {\map {\sigma_1} 6} = 2$

- 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 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.

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- 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

### $6$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 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 $6$ |

## 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

### Previous in Sequence: $1$

*Previous ... Next*: Hexagonal Number*Previous ... Next*: Pentagonal Pyramidal Number*Previous ... Next*: Ore Number*Previous ... Next*: Sequence of Powers of 6*Previous ... Next*: Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors

### Previous in Sequence: $2$

*Previous ... Next*: Special Highly Composite Number*Previous ... Next*: Central Binomial Coefficient*Previous ... Next*: Factorial*Previous ... Next*: Primorial*Previous ... Next*: Primorials which are Product of Consecutive Integers

### Previous in Sequence: $3$

*Previous ... Next*: Numbers not Sum of Distinct Squares*Previous ... Next*: Triangular Number*Previous ... Next*: Woodall Prime*Previous ... Next*: Palindromic Triangular Numbers*Previous ... Next*: Palindromic Triangular Numbers with Palindromic Index

### Previous in Sequence: $4$

*Previous ... Next*: Sequence of Integers whose Factorial minus 1 is Prime*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares

*Previous ... Next*: Ulam Number*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers*Previous ... Next*: Integers such that all Coprime and Less are Prime

*Previous ... Next*: Semiprime Number

### Previous in Sequence: $5$

*Previous ... Next*: Pluperfect Digital Invariant*Previous ... Next*: Zuckerman Number*Previous ... Next*: Harshad Number*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation*Previous ... Next*: Powers of 2 and 5 without Zeroes*Previous ... Next*: Numbers which are Sum of Increasing Powers of Digits*Previous ... Next*: Sum of Sequence of Alternating Positive and Negative Factorials being Prime*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum

*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Previous ... Next*: Trimorphic Number*Previous ... Next*: Alternating Groups that are Ambivalent*Previous ... Next*: Consecutive Integers whose Product is Primorial*Previous ... Next*: Automorphic Number

### Next in Sequence: $10$ and above

*Next*: Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways*Next*: Semiperfect Number*Next*: Primitive Semiperfect Number*Next*: Perfect Number*Next*: Unitary Perfect Number*Next*: Triangular Numbers which are Product of 3 Consecutive Integers*Next*: Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors

## 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

## Linguistic Note

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

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

## 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