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