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-non-zero number:
 * $\forall n: n \ne 0 \implies n^1 = 1$


 * 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 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 counting 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 value equals its cube root:
 * $\map {\sigma_0} 1 = 1$


 * 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

$2$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


 * 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