1

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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 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:
$2 \times 1^2 - 1 = 2 \times 1 - 1 = 1 = 1^2$


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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 Marin Mersenne 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 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


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 $1$:
$1 = \ds \dfrac 1 {1^2} \sum_{k \mathop = 1}^{1^3} k = \dfrac {1 \paren {1^3 + 1} } 2$


Arithmetic Functions on $1$

\(\ds \map {\sigma_0} { 1 }\) \(=\) \(\ds 1\) $\sigma_0$ of $1$
\(\ds \map \phi { 1 }\) \(=\) \(\ds 1\) $\phi$ of $1$
\(\ds \map {\sigma_1} { 1 }\) \(=\) \(\ds 1\) $\sigma_1$ of $1$


Also see


Previous in sequence: $0$

Next in sequence: $2$


Next in sequence: $3$


Next in sequence: $4$


Next in sequence: $5$


Next in sequence: $8$ and above


Previous in sequence: $2$


Next in sequence: $2$


Next in sequence: $3$


Next in sequence: $4$


Next in sequence: $5$


Next in sequence: $6$


Next in sequence: $7$


Next in sequence: $8$


Next in sequence: $9$


Next in sequence: $10$


Next in sequence: $11$


Next in sequence: $12$ and above


Historical Note

The ancient Greeks did not consider $1$ to be a number.

According to the Pythagoreans, the number One ($1$) was the Generator of all Numbers: the omnipotent One.

It represented reason, for reason could generate only $1$ self-evident body of truth.


While a number, according to Euclid, was an aggregate of units, a unit was not considered to be an aggregate of itself.

The much-quoted statement of Jakob Köbel might as well be repeated here:

Wherefrom thou understandest that $1$ is no number but it is a generatrix beginning and foundation for all other numbers.
-- $1537$

illustrating that this mindset still held sway as late as the $16$th century.


The ancient Greeks considered $1$ as both odd and even by fallacious reasoning.


Linguistic Note

Words derived from or associated with the number $1$ include:

From English one, from Germanic
  • alone: literally: all one
  • only: literally: one-like
  • lonely: a combination of alone and only, hence all one-like
  • atone: to repent of one's misdeeds by becoming at one with the person you have offended against
  • none: literally not one
  • once: meaning one time
  • nonce, as in for the nonce, or nonce word: meaning just for this one time
From Latin unum
  • united: many things that have become one
  • union: an arrangement where the members are unified into one body
  • unanimous: where people speak with one spirit
  • universal: as in universal principle, for example: holding throughout the universe
  • university: students and professors are turned into one body
  • uniform: clothes all of one form
  • unicorn: a mythical beast with one horn
  • onion: from the same root as union: the Romans referred to it as one large pearl
From Greek μόνος (mónos)
  • monologue: a speech by $1$ person
  • monopoly: selling by only $1$ agency
  • monk: from the Greek monakos: someone who is alone or solitary
  • monolith: something made from one stone
  • monogram: a way to write your name with just one drawing of the pen
  • monotonous: all of one (boring) type


Sources

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