1

Number
$1$ (one) is:


 * The smallest (strictly) positive integer


 * The only (strictly) positive integer which is neither prime nor composite


 * The $1$st triangular number:
 * $1 = \dfrac {1 \paren {1 + 1} } 2$


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


 * The $1$st highly composite number:
 * $\tau \paren 1 = 1$


 * The $1$st special highly composite number


 * The $1$st highly abundant number:
 * $\sigma \paren 1 = 1$


 * The $1$st superabundant number:
 * $\dfrac {\sigma \paren 1} 1 = \dfrac 1 1 = 1$


 * The zeroth power of every non-non-zero number:
 * $\forall n: n \ne 0 \implies n^1 = 1$


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


 * The $1$st factorial:
 * $1 = 1!$


 * The $1$st superfactorial:
 * $1 = 1\$ = 1!$


 * 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 $m! + 1$ (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 $1$st and $2$nd Fibonacci numbers after the zeroth ($0$):
 * $1 = 0 + 1$


 * The $1$st Lucas number after the zeroth $(2)$


 * The $0$th and $1$st Catalan numbers:
 * $\dfrac 1 {0 + 1} \dbinom {2 \times 0} 1 = \dfrac 1 1 \times 1 = 1$
 * $\dfrac 1 {1 + 1} \dbinom {2 \times 1} 1 = \dfrac 1 2 \times 2 = 1$


 * The $1$st Ulam number


 * 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 both $2^n$ and $5^n$ have no zeroes:
 * $2^1 = 2, 5^1 = 5$


 * The $1$st (strictly) positive integer which cannot be expressed as the sum of exactly $5$ non-zero squares


 * The $0$th term of Göbel's sequence, by definition


 * The $0$th term of the $3$-Göbel sequence, by definition


 * The $1$st of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes


 * 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 $1$st of the $5$ known powers of $2$ whose digits are also all powers of $2$


 * 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 $1$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 $2$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 $0$th and $1$st Bell numbers


 * 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:
 * $\sigma \paren 1 = 1 = 1^2$


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


 * 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 palindromic integer which is the index of a palindromic triangular number after $0$:
 * $T_1 = 1$

Also see

 * One is not Prime
 * Divisors of One

Next in sequence: $3$


[[Category:1]]