6
{{NumberPageLink|prev = 5|next = 7}} == 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. {{WIP|Set up the above as a sequence in its own right}} :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: ::$6 = \dfrac {3 \times 4} 2 = \dfrac {3 + 4 + 5} 2$ {{ArithmeticFunctionTable|n = 6|sigma = 12|tau = 4|phi = 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 === Previous in Sequence: $1$ === * {{NumberPageLink|prev = 1|next = 15|type = Hexagonal Number|cat = Hexagonal Numbers}} * {{NumberPageLink|prev = 1|next = 18|type = Pentagonal Pyramidal Number|cat = Pyramidal Numbers}} * {{NumberPageLink|prev = 1|next = 28|type = Ore Number|cat = Ore Numbers}} * {{NumberPageLink|prev = 1|next = 36|result = Sequence of Powers of 6|cat = Powers of 6}} * {{NumberPageLink|prev = 1|next = 140|result = Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors}} === Previous in Sequence: $2$ === * {{NumberPageLink|prev = 2|next = 12|type = Special Highly Composite Number|cat = Special Highly Composite Numbers}} * {{NumberPageLink|prev = 2|next = 20|type = Central Binomial Coefficient|cat = Central Binomial Coefficients}} * {{NumberPageLink|prev = 2|next = 24|type = Factorial|cat = Factorials}} * {{NumberPageLink|prev = 2|next = 30|type = Primorial|cat = Primorials}} * {{NumberPageLink|prev = 2|next = 30|result = Primorials which are Product of Consecutive Integers}} === Previous in Sequence: $3$ === * {{NumberPageLink|prev = 3|next = 7|result = Numbers not Sum of Distinct Squares}} * {{NumberPageLink|prev = 3|next = 10|type = Triangular Number|cat = Triangular Numbers}} * {{NumberPageLink|prev = 3|next = 30|type = Woodall Prime|cat = Woodall Primes}} * {{NumberPageLink|prev = 3|next = 55|result = Palindromic Triangular Numbers}} * {{NumberPageLink|prev = 3|next = 66|result = Palindromic Triangular Numbers with Palindromic Index}} === Previous in Sequence: $4$ === * {{NumberPageLink|prev = 4|next = 7|result = Sequence of Integers whose Factorial minus 1 is Prime}} * {{NumberPageLink|prev = 4|next = 7|result = Integer not Expressible as Sum of 5 Non-Zero Squares}} * {{NumberPageLink|prev = 4|next = 8|type = Ulam Number|cat = Ulam Numbers}} * {{NumberPageLink|prev = 4|next = 8|type = Highly Abundant Number|cat = Highly Abundant Numbers}} * {{NumberPageLink|prev = 4|next = 8|result = Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers}} * {{NumberPageLink|prev = 4|next = 8|result = Integers such that all Coprime and Less are Prime}} * {{NumberPageLink|prev = 4|next = 9|type = Semiprime Number|cat = Semiprimes}} * {{NumberPageLink|prev = 4|next = 12|type = Highly Composite Number|cat = Highly Composite Numbers}} * {{NumberPageLink|prev = 4|next = 12|type = Superabundant Number|cat = Superabundant Numbers}} === Previous in Sequence: $5$ === * {{NumberPageLink|prev = 5|next = 7|type = Pluperfect Digital Invariant|cat = Pluperfect Digital Invariants}} * {{NumberPageLink|prev = 5|next = 7|type = Zuckerman Number|cat = Zuckerman Numbers}} * {{NumberPageLink|prev = 5|next = 7|type = Harshad Number|cat = Harshad Numbers}} * {{NumberPageLink|prev = 5|next = 7|result = Powers of 2 with no Zero in Decimal Representation}} * {{NumberPageLink|prev = 5|next = 7|result = Powers of 5 with no Zero in Decimal Representation}} * {{NumberPageLink|prev = 5|next = 7|result = Powers of 2 and 5 without Zeroes}} * {{NumberPageLink|prev = 5|next = 7|result = Numbers which are Sum of Increasing Powers of Digits}} * {{NumberPageLink|prev = 5|next = 7|result = Sum of Sequence of Alternating Positive and Negative Factorials being Prime}} * {{NumberPageLink|prev = 5|next = 8|result = Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes}} * {{NumberPageLink|prev = 5|next = 9|type = Trimorphic Number|cat = Trimorphic Numbers}} * {{NumberPageLink|prev = 5|next = 14|result = Consecutive Integers whose Product is Primorial}} * {{NumberPageLink|prev = 5|next = 25|type = Automorphic Number|cat = Automorphic Numbers}} === Next in Sequence: $10$ and above === * {{NumberPageLink|next = 10|result = Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways}} * {{NumberPageLink|next = 12|type = Semiperfect Number|cat = Primitive Semiperfect Numbers}} * {{NumberPageLink|next = 20|type = Primitive Semiperfect Number|cat = Semiperfect Numbers}} * {{NumberPageLink|next = 28|type = Perfect Number|cat = Perfect Numbers}} * {{NumberPageLink|next = 60|type = Unitary Perfect Number|cat = Unitary Perfect Numbers}} * {{NumberPageLink|next = 120|result = Triangular Numbers which are Product of 3 Consecutive Integers}} * {{NumberPageLink|next = 140|result = Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors}} == Historical Note == {{:6/Historical Note}} == Linguistic Note == {{:6/Linguistic Note}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Definition:Automorphic Number/Sequence|next = Prime equals Plus or Minus One modulo 6}}: $5$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Volume of Unit Hypersphere/Sequence|next = 6/Historical Note}}: $6$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Definition:Automorphic Number/Sequence|next = Prime equals Plus or Minus One modulo 6}}: $5$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Volume of Unit Hypersphere/Sequence|next = 6/Historical Note}}: $6$