1

== Number ==
$1$ (one) is:
:The immediate successor element of zero in the set of natural numbers $\N$
:The only (strictly) positive integer which is neither prime nor composite
:The only (strictly) positive integer which is a divisor of every integer
=== $0$th Term ===
:The $0$th (zeroth) power of every non-zero number:
::$\forall n: n \ne 0 \implies n^0 = 1$
:The $0$th term of Göbel's sequence, by definition
:The $0$th term of the $3$-Göbel sequence, by definition
:The $0$th and $1$st Catalan numbers:
::$1 = \dfrac 1 {0 + 1} \dbinom {2 \times 0} 1 = \dfrac 1 1 \times 1$
::$1 = \dfrac 1 {1 + 1} \dbinom {2 \times 1} 1 = \dfrac 1 2 \times 2$
:The $0$th and $1$st Bell numbers
:The smallest positive integer the decimal expansion of whose reciprocal has a period of $0$:
::$\dfrac 1 1 = 1 \cdotp 0$
=== $1$st Term ===
:The $1$st (strictly) positive integer
:The $1$st square number:
::$1 = 1^2$
:The $1$st triangular number:
::$1 = \dfrac {1 \times \paren {1 + 1} } 2$
:The $1$st (positive) odd number
::$1 = 0 \times 2 + 1$
:The $1$st number to be both square and triangular:
::$1 = 1^2 = \dfrac {1 \times \paren {1 + 1}} 2$
:The $1$st square which has no more than $2$ distinct digits and does not end in $0$:
::$1 = 1^2$
:The $1$st square number to be the divisor sum of some (strictly) positive integer:
::$1 = \map {\sigma_1} 1$
:The $1$st generalized pentagonal number:
::$1 = \dfrac {1 \paren {3 \times 1 - 1} } 2$
:The $1$st highly composite number:
::$\map {\sigma_0} 1 = 1$
:The $1$st special highly composite number
:The $1$st highly abundant number:
::$\map {\sigma_1} 1 = 1$
:The $1$st superabundant number:
::$\dfrac {\map {\sigma_1} 1} 1 = \dfrac 1 1 = 1$
:The $1$st almost perfect number:
::$\map {\sigma_1} 1 = 1 = 2 - 1$
:The $1$st factorial:
::$1 = 1!$
:The $1$st superfactorial:
::$1 = 1\$ = 1!$
:The $1$st Lucas number after the zeroth $(2)$
:The $1$st Ulam number
:The $1$st (strictly) positive integer which cannot be expressed as the sum of exactly $5$ non-zero squares
:The $1$st of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes
:The $1$st of the $5$ known powers of $2$ whose digits are also all powers of $2$
:The $1$st factorion base $10$:
::$1 = 1!$
:The $1$st of the trivial $1$-digit pluperfect digital invariants:
::$1^1 = 1$
:The $1$st of the $1$st pair of consecutive integers whose product is a primorial:
::$1 \times 2 = 2 = 2 \#$
:The $1$st of the (trivial $1$-digit) Zuckerman numbers:
::$1 = 1 \times 1$
:The $1$st of the (trivial $1$-digit) harshad numbers:
::$1 = 1 \times 1$
:The $1$st positive integer whose cube is palindromic (in this case trivially):
::$1^3 = 1$
:The $1$st lucky number
:The $1$st palindromic lucky number
:The $1$st Stern number
:The $1$st Stern prime
:The $1$st Cullen number:
::$1 = 0 \times 2^0 + 1$
:The $1$st number whose divisor sum is square:
::$\map {\sigma_1} 1 = 1 = 1^2$
:The $1$st positive integer after $1$ of which the product of its Euler $\phi$ function and its divisor count function equals its divisor sum:
::$\map \phi 1 \map {\sigma_0} 1 = 1 \times 1 = 1 = \map {\sigma_1} 1$
:The $1$st positive integer solution to $\map \phi n = \map \phi {n + 1}$:
::$\map \phi 1 = 1 = \map \phi 2$
:The $1$st element of the Fermat set
:The $1$st integer $n$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} 1 = 1$, $\map \phi 1 = 1$, $\map {\sigma_1} 1 = 1$
:The $1$st Lucas number which is also triangular
:The $1$st tetrahedral number:
::$1 = \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6$
:The $1$st of the $3$ tetrahedral numbers which are also square
:The $1$st trimorphic number:
::$1^3 = \mathbf 1$
:The $1$st powerful number (vacuously)
:The $1$st integer which equals the number of digits in its factorial:
::$1! = 1$
:which has $1$ digit
:The $1$st power of $2$ which is the sum of distinct powers of $3$:
::$1 = 2^0 = 3^0$
:The $1$st square which has no more than $2$ distinct digits
:The $1$st pentagonal number:
::$1 = \dfrac {1 \paren {1 \times 3 - 1} } 2$
:The $1$st pentagonal number which is also palindromic:
::$1 = \dfrac {1 \paren {1 \times 3 - 1} } 2$
:The $1$st square pyramidal number:
::$1 = \dfrac {1 \paren {1 + 1} \paren {2 \times 1 + 1} } 6$
:The $1$st pentatope number:
::$1 = \dfrac {1 \paren {1 + 1} \paren {1 + 2} \paren {1 + 3} } {24}$
:The $1$st automorphic number:
::$1^2 = \mathbf 1$
:The $1$st number such that $2 n^2 - 1$ is square:
::$2 \times 1^2 - 1 = 2 \times 1 - 1 = 1 = 1^2$
:The $1$st Ore number:
::$\dfrac {1 \times \map {\sigma_0} 1} {\map {\sigma_1} 1} = 1$
:and the $1$st whose divisors also have an arithmetic mean which is an integer:
::$\dfrac {\map {\sigma_1} 1} {\map {\sigma_0} 1} = 1$
:The $1$st hexagonal number:
::$1 = 1 \paren {2 \times 1 - 1}$
:The $1$st pentagonal pyramidal number:
::$1 = \dfrac {1^2 \paren {1 + 1} } 2$
:The $1$st heptagonal number:
::$1 = \dfrac {1 \paren {5 \times 1 - 3} } 2$
:The $1$st centered hexagonal number:
::$1 = 1^3 - 0^3$
:The $1$st hexagonal pyramidal number:
:The $1$st Woodall number:
::$1 = 1 \times 2^1 - 1$
:The $1$st happy number:
::$1 \to 1^2 = 1$
:The $1$st positive integer the divisor sum is a cube:
::$\map {\sigma_1} 1 = 1 = 1^3$
:The $1$st cube number:
::$1 = 1^3$
:The $1$st of the only two cubic Fibonacci numbers
:The $1$st octagonal number:
::$1 = 1 \paren {3 \times 1 - 2}$
:The $1$st heptagonal pyramidal number:
::$1 = \dfrac {1 \paren {1 + 1} \paren {5 \times 1 - 2} } 6$
:The $1$st Kaprekar triple:
::$1^3 = 1 \to 0 + 0 + 1 = 1$
:The $1$st palindromic cube:
::$1 = 1^3$
:The $1$st Kaprekar number:
::$1^2 = 01 \to 0 + 1 = 1$
:The $1$st number whose square has a divisor sum which is itself square:
::$\map {\sigma_1} 1 = 1 = 1^2$
:The $1$st of the $5$ tetrahedral numbers which are also triangular
:The $1$st positive integer which cannot be expressed as the sum of a square and a prime
:The $1$st positive integer such that all smaller positive integers coprime to it are prime
:The (trivial) $1$st repunit
:The $1$st fourth power:
::$1 = 1 \times 1 \times 1 \times 1$
:The $1$st integer $m$ whose cube can be expressed (trivially) as the sum of $m$ consecutive squares:
::$1^3 = \ds \sum_{k \mathop = 1}^1 \paren {0 + k}^2$
:The $1$st and $2$nd Fibonacci numbers after the zeroth ($0$):
::$1 = 0 + 1$
:The $1$st positive integer whose divisor sum of its Euler $\phi$ value equals its divisor sum:
::$\map {\sigma_1} {\map \phi 1} = \map {\sigma_1} 1 = 1 = \map {\sigma_1} 1$
:The $1$st square pyramorphic number:
::$1 = \ds \sum_{k \mathop = 1}^1 k^2 = \dfrac {1 \paren {1 + 1} \paren {2 \times 1 + 1} } 6$
:The $1$st of the $4$ square pyramidal numbers which are also triangular
:The $1$st Wonderful Demlo number
:The $1$st obstinate number
:The index of the $1$st Cullen prime:
::$1 \times 2^1 + 1 = 3$
:The total number of permutations of $r$ objects from a set of $1$ object, where $1 \le r \le 1$
:The index of the $1$st Mersenne number which asserted to be prime
::($1$ itself was classified as a prime number in those days)
:The number of different representations of $1$ as the sum of $1$ unit fractions (degenerate case)
:The $1$st centered hexagonal number which is also square
:The $1$st pentagonal number which is also triangular:
::$1 = \dfrac {1 \paren {3 \times 1 - 1} } 2 = \dfrac {1 \times \paren {1 + 1} } 2$
:The $1$st odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
:The $1$st odd number which cannot be expressed as the sum of an integer power and a prime number
:The $1$st (strictly) positive integer which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The $1$st of $6$ integers $n$ such that the alternating group $A_n$ is ambivalent
:The $1$st of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
:The $1$st integer whose divisor sum divided by its Euler $\phi$ value is a square:
::$\dfrac {\map {\sigma_1} 1} {\map \phi 1} = \dfrac 1 1 = 1 = 1^2$
:The $1$st positive integer whose Euler $\phi$ value is equal to the product of its digits: 
::$\map \phi 1 = 1$
:The $1$st of the $3$ positive integers whose divisor count equals its cube root:
::$\map {\sigma_0} 1 = 1$
:The $1$st positive integer solution to the Ramanujan-Nagell equation $x^2 + 7 = 2^n$ for integral $n$:
::$1^2 + 7 = 8 = 2^3$
:The $1$st self-locating number in $\pi$:
::$3 \cdotp \boldsymbol 1459 \ldots$
:The number of distinct free monominoes
:The number of different binary operations that can be applied to a set with $1$ element
:The number of different binary operations with an identity element that can be applied to a set with $1$ element
:The number of different commutative binary operations that can be applied to a set with $1$ element
:The number of integer partitions for $1$:
::$\map p 1 = 1$
=== $2$nd Term ===
:The $2$nd after $0$ of the $5$ Fibonacci numbers which are also triangular
:The $2$nd palindromic triangular number after $0$
:The $2$nd integer $n$ after $0$ such that $2^n$ contains no zero in its decimal representation:
::$2^1 = 2$
:The $2$nd integer $n$ after $0$ such that $5^n$ contains no zero in its decimal representation:
::$5^1 = 5$
:The $2$nd integer $n$ after $0$ such that both $2^n$ and $5^n$ have no zeroes:
::$2^1 = 2, 5^1 = 5$
:The $2$nd palindromic integer which is the index of a palindromic triangular number after $0$:
::$T_1 = 1$
:The $1$st palindromic integer after $0$ whose square is also palindromic integer
::$1^2 = 1$
:The $2$nd Dudeney number after $0$:
::$1^3 = 1$
:The $2$nd number after $0$ which is (trivially) the sum of the increasing powers of its digits taken in order:
::$1^1 = 1$
:The $2$nd non-negative integer $n$ after $0$ such that the Fibonacci number $F_n$ ends in $n$
:The $2$nd after $0$ of the $3$ Fibonacci numbers which equals its index
::$F_1 = 1$
:The $2$nd subfactorial after $0$:
::$1 = 2! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} }$
:The $2$nd integer $m$ after $0$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
::$1^2 = \dbinom 0 0 + \dbinom 0 1 + \dbinom 0 2 + \dbinom 0 3$
:The $2$nd integer $m$ after $0$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$1! + 1 = 1 + 1 = 2$
:The $2$nd integer after $0$ such that its double factorial plus $1$ is prime:
::$1!! + 1 = 2$
:The $2$nd integer after $0$ which is palindromic in both decimal and binary:
::$1_{10} = 1_2$
:The $2$nd integer after $0$ which is palindromic in both decimal and ternary:
::$1_{10} = 1_3$
:The $2$nd Ramanujan-Nagell number after $0$:
::$1 = 2^1 - 1 = \dfrac {1 \paren {1 + 1} } 2$
:The number of different representations of $1$ as the sum of $2$ unit fractions:
::$1 = \dfrac 1 2 + \dfrac 1 2$
:The number of distinct free dominoes
=== Miscellaneous ===
:The total of all the entries in the trivial magic square of order $1$:
::$1 = \ds \sum_{k \mathop = 1}^{1^2} k = \dfrac {1^2 \paren {1^2 + 1} } 2$
:The total of all the entries in the trivial magic cube of order $1$:
::$1 = \ds \sum_{k \mathop = 1}^{1^3} k = \dfrac {1^3 \paren {1^3 + 1} } 2$
:The magic constant of the trivial magic square of order $1$:
::$1 = \ds \dfrac 1 1 \sum_{k \mathop = 1}^{1^2} k = \dfrac {1 \paren {1^2 + 1} } 2$
:The magic constant of the trivial magic cube of order $1$:
::$1 = \ds \dfrac 1 {1^2} \sum_{k \mathop = 1}^{1^3} k = \dfrac {1 \paren {1^3 + 1} } 2$
== Also see ==
* Definition:Unity
* One is not Prime
* Divisors of One
* One Divides all Integers
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== Historical Note ==
== Linguistic Note ==
== Sources ==
* : $1$
* 
* : Cool Numbers: Self-location and $\pi$
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Category:Specific Numbers
Category:1