# 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

### $1$st Term

- The $1$st (strictly) positive integer

- The $1$st (strictly) positive integer

- 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 number to be the $\sigma$ (sigma) value of some (strictly) positive integer:
- $1 = \map \sigma 1$

- The $1$st generalized pentagonal number:
- $1 = \dfrac {1 \paren {3 \times 1 - 1} } 2$

- The $1$st highly composite number:
- $\map \tau 1 = 1$

- The $1$st special highly composite number

- The $1$st highly abundant number:
- $\map \sigma 1 = 1$

- The $1$st superabundant number:
- $\dfrac {\map \sigma 1} 1 = \dfrac 1 1 = 1$

- The $1$st almost perfect number:
- $\map \sigma 1 = 1 = 2 - 1$

- 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 $\sigma$ value is square:
- $\map \sigma 1 = 1 = 1^2$

- The $1$st positive integer after $1$ of which the product of its Euler $\phi$ function and its $\tau$ function equals its $\sigma$ function:
- $\map \phi 1 \map \tau 1 = 1 \times 1 = 1 = \map \sigma 1$

- 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 $\map \tau n \divides \map \phi n \divides \map \sigma n$:
- $\map \tau 1 = 1$, $\map \phi 1 = 1$, $\map \sigma 1 = 1$

- The $1$st Lucas number which is also triangular

- The $1$st tetrahedral number:
- $1 = \dfrac {1 \left({1 + 1}\right) \left({1 + 2}\right)} 6$

- 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 pentagonal number:
- $1 = \dfrac {1 \left({1 \times 3 - 1}\right)} 2$

- The $1$st pentagonal number which is also palindromic:
- $1 = \dfrac {1 \left({1 \times 3 - 1}\right)} 2$

- 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:
- $\dfrac {1 \times \map \tau 1} {\map \sigma 1} = 1$

- and the $1$st whose divisors also have an arithmetic mean which is an integer:
- $\dfrac {\map \sigma 1} {\map \tau 1} = 1$

- 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 sum of whose divisors is a cube:
- $\map \sigma 1 = 1 = 1^3$

- 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 $\sigma$ value which is itself square:
- $\map \sigma 1 = 1 = 1^2$

- 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:
- $1^3 = \displaystyle \sum_{k \mathop = 1}^1 \left({0 + k}\right)^2$

- The $1$st and $2$nd Fibonacci numbers after the zeroth ($0$):
- $1 = 0 + 1$

- The $1$st positive integer whose $\sigma$ value of its Euler $\phi$ value equals its $\sigma$ value:
- $\map \sigma {\map \phi 1} = \map \sigma 1 = 1 = \map \sigma 1$

- The $1$st square pyramorphic number:
- $1 = \displaystyle \sum_{k \mathop = 1}^1 k^2 = \dfrac {1 \paren {1 + 1} \paren {2 \times 1 + 1} } 6$

- 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 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 number of distinct free monominoes

### $2$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 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 $0$:
- $1 = 2^1 - 1 = \dfrac {1 \left({1 + 1}\right)} 2$

- The number of different representations of $1$ as the sum of $2$ unit fractions:
- $1 = \dfrac 1 2 + \dfrac 1 2$

### Miscellaneous

- The total of all the entries in the trivial magic square of order $1$:
- $1 = \displaystyle \sum_{k \mathop = 1}^{1^2} k = \dfrac {1^2 \paren {1^2 + 1} } 2$

- The total of all the entries in the trivial magic cube of order $1$:
- $1 = \displaystyle \sum_{k \mathop = 1}^{1^3} k = \dfrac {1^3 \paren {1^3 + 1} } 2$

- The magic constant of the trivial magic square of order $1$:
- $1 = \displaystyle \dfrac 1 1 \sum_{k \mathop = 1}^{1^2} k = \dfrac {1 \paren {1^2 + 1} } 2$

- The magic constant of the trivial magic cube of order $1$:
- $1 = \displaystyle \dfrac 1 {1^2} \sum_{k \mathop = 1}^{1^3} k = \dfrac {1 \paren {1^3 + 1} } 2$

### Arithmetic Functions on $1$

\(\ds \map \tau { 1 }\) | \(=\) | \(\ds 1\) | $\tau$ of $1$ | |||||||||||

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

\(\ds \map \sigma { 1 }\) | \(=\) | \(\ds 1\) | $\sigma$ of $1$ |

## Also see

### Previous in sequence: $0$

#### Next in sequence: $2$

*Previous ... Next*: Subfactorial*Previous ... Next*: Sequence of Integers whose Factorial plus 1 is Prime*Previous ... Next*: Prime Values of Double Factorial plus 1*Previous ... Next*: Fibonacci Number*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation*Previous ... Next*: Powers of 2 and 5 without Zeroes*Previous ... Next*: Palindromes in Base 10 and Base 3*Previous ... Next*: Numbers which are Sum of Increasing Powers of Digits*Previous ... Next*: Square Formed from Sum of 4 Consecutive Binomial Coefficients*Previous ... Next*: Palindromic Indices of Palindromic Triangular Numbers*Previous ... Next*: Square of Small-Digit Palindromic Number is Palindromic

#### Next in sequence: $3$

*Previous ... Next*: Triangular Number*Previous ... Next*: Palindromes in Base 10 and Base 2*Previous ... Next*: Triangular Fibonacci Numbers*Previous ... Next*: Palindromic Triangular Numbers*Previous ... Next*: Ramanujan-Nagell Number

#### Next in sequence: $5$

*Previous ... Next*: Fibonacci Numbers which equal their Index*Previous ... Next*: Sequence of Fibonacci Numbers ending in Index

#### Next in sequence: $8$

*Previous ... Next*: Dudeney Number

### Previous in sequence: $2$

*Previous ... Next*: Lucas Number

### Next in sequence: $2$

*Next*: Generalized Pentagonal Number*Next*: Highly Composite Number*Next*: Special Highly Composite Number*Next*: Highly Abundant Number*Next*: Superabundant Number*Next*: Almost Perfect Number*Next*: Factorial*Next*: Superfactorial*Next*: Fibonacci Number*Next*: Catalan Number*Next*: Ulam Number*Next*: Sequence of Powers of 2*Next*: Powers of 2 with no Zero in Decimal Representation*Next*: Powers of 2 and 5 without Zeroes*Next*: Integer not Expressible as Sum of 5 Non-Zero Squares*Next*: Integers such that all Coprime and Less are Prime*Next*: Göbel's Sequence*Next*: 3-Göbel Sequence*Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Next*: Powers of 2 whose Digits are Powers of 2*Next*: Pluperfect Digital Invariant*Next*: Factorions Base 10*Next*: Consecutive Integers whose Product is Primorial*Next*: Zuckerman Number*Next*: Harshad Number*Next*: Square Formed from Sum of 4 Consecutive Binomial Coefficients*Next*: Bell Number*Next*: Sequence of Integers whose Cube is Palindromic*Next*: Number of Free Polyominoes*Next*: Mersenne Prime/Historical Note

### Next in sequence: $3$

*Next*: Sequence of Powers of 3*Next*: Lucky Number*Next*: Sequence of Palindromic Lucky Numbers*Next*: Stern Number*Next*: Cullen Number*Next*: Numbers whose Sigma is Square*Next*: Integers whose Phi times Tau equal Sigma*Next*: Consecutive Integers with Same Euler Phi Value*Next*: Fermat Set*Next*: Numbers such that Tau divides Phi divides Sigma*Next*: Triangular Lucas Numbers*Next*: Representation of 1 as Sum of n Unit Fractions*Next*: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares

### Next in sequence: $4$

*Next*: Sequence of Powers of 4*Next*: Tetrahedral Number*Next*: Square and Tetrahedral Numbers*Next*: Trimorphic Number*Next*: Powerful Number*Next*: Powers of 2 which are Sum of Distinct Powers of 3*Next*: Squares with No More than 2 Distinct Digits

### Next in sequence: $5$

*Next*: Sequence of Powers of 5*Next*: Pentagonal Number*Next*: Sequence of Palindromic Pentagonal Numbers*Next*: Square Pyramidal Number*Next*: Pentatope Number*Next*: Automorphic Number*Next*: Magic Constant of Magic Square*Next*: Odd Numbers not Sum of Prime and Power

### Next in sequence: $6$

*Next*: Sequence of Powers of 6*Next*: Ore Number*Next*: Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors*Next*: Hexagonal Number*Next*: Pentagonal Pyramidal Number

### Next in sequence: $7$

*Next*: Sequence of Powers of 7*Next*: Centered Hexagonal Number*Next*: Hexagonal Pyramidal Number*Next*: Heptagonal Number*Next*: Woodall Number*Next*: Happy Number*Next*: Integers whose Sigma Value is Cube

### Next in sequence: $8$

*Next*: Sequence of Powers of 8*Next*: Cube Number*Next*: Cubic Fibonacci Numbers*Next*: Octagonal Number*Next*: Heptagonal Pyramidal Number*Next*: Kaprekar Triple*Next*: Sequence of Palindromic Cubes

### Next in sequence: $9$

*Next*: Sequence of Powers of 9*Next*: Kaprekar Number*Next*: Square Numbers whose Sigma is Square*Next*: Magic Constant of Magic Cube

### Next in sequence: $10$

*Next*: Sum of Terms of Magic Square*Next*: Sequence of Powers of 10*Next*: Tetrahedral and Triangular Numbers*Next*: Numbers not Sum of Square and Prime

### Next in sequence: $11$

### Next in sequence: $12$ and above

*Next*: Sequence of Powers of 12*Next*: Sequence of Powers of 13*Next*: Sequence of Powers of 14*Next*: Sequence of Powers of 15*Next*: Fourth Power*Next*: Sequence of Powers of 16*Next*: Numbers Equal to Number of Digits in Factorial*Next*: Integer both Square and Triangular*Next*: Sum of Terms of Magic Cube*Next*: Numbers whose Cube equals Sum of Sequence of that many Squares*Next*: Square Pyramidal and Triangular Numbers*Next*: Square Pyramorphic Number*Next*: Integers for which Sigma of Phi equals Sigma*Next*: Wonderful Demlo Number*Next*: Obstinate Number*Next*: Cullen Prime*Next*: Sequence of Square Centered Hexagonal Numbers*Next*: Triangular Numbers which are also Pentagonal

## 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:

**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**

**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**

**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

- 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $1$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**unity** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**unity** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**unity**

- Work To Do
- Subfactorials/Examples
- Fibonacci Numbers/Examples
- Triangular Numbers/Examples
- Ramanujan-Nagell Numbers/Examples
- Dudeney Numbers/Examples
- Lucas Numbers/Examples
- Generalized Pentagonal Numbers/Examples
- Highly Composite Numbers/Examples
- Special Highly Composite Numbers/Examples
- Highly Abundant Numbers/Examples
- Superabundant Numbers/Examples
- Almost Perfect Numbers/Examples
- Factorials/Examples
- Superfactorials/Examples
- Catalan Numbers/Examples
- Ulam Numbers/Examples
- Powers of 2/Examples
- Göbel's Sequence/Examples
- Pluperfect Digital Invariants/Examples
- Factorions/Examples
- Zuckerman Numbers/Examples
- Harshad Numbers/Examples
- Bell Numbers/Examples
- Mersenne's Assertion/Examples
- Powers of 3/Examples
- Lucky Numbers/Examples
- Stern Numbers/Examples
- Cullen Numbers/Examples
- Numbers whose Sigma is Square/Examples
- Powers of 4/Examples
- Tetrahedral Numbers/Examples
- Trimorphic Numbers/Examples
- Powerful Numbers/Examples
- Powers of 5/Examples
- Pentagonal Numbers/Examples
- Pyramidal Numbers/Examples
- Pentatope Numbers/Examples
- Automorphic Numbers/Examples
- Powers of 6/Examples
- Ore Numbers/Examples
- Hexagonal Numbers/Examples
- Powers of 7/Examples
- Centered Hexagonal Numbers/Examples
- Heptagonal Numbers/Examples
- Woodall Numbers/Examples
- Happy Numbers/Examples
- Powers of 8/Examples
- Cube Numbers/Examples
- Octagonal Numbers/Examples
- Kaprekar Numbers/Examples
- Powers of 9/Examples
- Square Numbers whose Sigma is Square/Examples
- Powers of 10/Examples
- Powers of 11/Examples
- Repunits/Examples
- Powers of 12/Examples
- Powers of 13/Examples
- Powers of 14/Examples
- Powers of 15/Examples
- Fourth Powers/Examples
- Powers of 16/Examples
- Numbers whose Cube equals Sum of Sequence of that many Squares/Examples
- Square Pyramorphic Numbers/Examples
- Wonderful Demlo Numbers/Examples
- Obstinate Numbers/Examples
- Cullen Primes/Examples
- Specific Numbers
- 1