# 2

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## Number

$2$ (two) is:

The $1$st prime number

The successor integer to the number $1$ (one).

The only even prime number

### Zeroth Term

The $0$th (zeroth) Lucas number

The $0$th Euclid number:
$2 = p_0\# + 1 = 1 + 1$

The $0$th Thabit number, and $1$st Thabit prime:
$2 = 3 \times 2^0 - 1$

### $1$st Term

The $1$st (strictly) positive even number

The $1$st (trivial, $1$-digit) palindromic prime

The $1$st Sophie Germain prime:
$2 \times 2 + 1 = 5$, which is prime

The index of the $1$st repunit prime:
$R_2 = 11$

The index of the $1$st Mersenne prime:
$M_2 = 2^2 - 1 = 3$

The index of the $1$st Woodall prime:
$2 \times 2^2 - 1 = 7$

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

The $1$st of the $4$ known primes $p$ such that $\dfrac {p^p - 1} {p - 1}$ is itself prime:
$\dfrac {2^2 - 1} {2 - 1} = 3$

The $1$st power of $2$ after the zeroth $1$:
$2^1 = 2$

The $1$st Fibonacci prime

The $1$st Lucas prime

The $1$st permutable prime

The $1$st untouchable number

The $1$st primorial:
$2 = p_1 \# = 2 \# := \displaystyle \prod_{k \mathop = 1}^1 p_k = 2$

The $1$st central binomial coefficient:
$2 = \dbinom {2 \times 1} 1 := \dfrac {2!} {\paren {1!}^2}$

The $1$st prime number of the form $n! + 1$ for integer $n$:
$1! + 1 = 1 + 1 = 2$
where $n!$ denotes $n$ factorial

The $1$st prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
$2 \# + 1 = 2 + 1 = 3$

The $1$st of the lucky numbers of Euler:
$n^2 + n + 2$ is prime for $n = 0$

The $1$st term of Göbel's sequence after the $0$th term $1$:
$2 = \paren {1 + 1^2} / 1$

The $1$st term of the $3$-Göbel sequence after the $0$th term $1$:
$2 = \paren {1 + 1^3} / 1$

The $1$st Sierpiński number of the first kind:
$2 = 1^1 + 1$

The $1$st prime Sierpiński number of the first kind:
$2 = 1^1 + 1$

The $1$st even number which cannot be expressed as the sum of $2$ composite odd numbers

The $1$st prime number which divides the sum of all smaller primes:
$0 = 0 \times 2$
(there are no primes smaller than $2$)

The $1$st even integer that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes

The $1$st positive integer which is not the sum of $1$ or more distinct squares

The $1$st positive integer which cannot be expressed as the sum of distinct pentagonal numbers

The $1$st prime number of the form $n^2 + 1$:
$2 = 1^2 + 1$

The $1$st of $4$ numbers whose letters, when spelt in French, are in alphabetical order:
deux

The $1$st integer at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied

The $1$st primorial which can be expressed as the product of consecutive integers:
$2 \# = 2 = 1 \times 2$

The $1$st of the $2$nd pair of consecutive integers whose product is a primorial:
$2 \times 3 = 6 = 3 \#$

The $1$st tri-automorphic number:
$2^2 \times 3 = 1 \mathbf 2$

The $1$st Euclid prime:
$2 = p_0\# + 1 = 1 + 1$

The $1$st (trivially) two-sided prime

The $1$st prime number consisting (trivially) of a string of consecutive ascending digits

The $1$st Hardy-Ramanujan number: the smallest positive integer which can be expressed as the sum of $2$ cubes in (trivially) $1$ way:
$2 = \map {\operatorname {Ta} } 1 = 1^3 + 1^3$

The $1$st of $6$ integers which cannot be expressed as the sum of distinct triangular numbers

### $2$nd Term

The $2$nd Catalan number after $(1)$, $1$:
$2 = \dfrac 1 {2 + 1} \dbinom {2 \times 2} 2 = \dfrac 1 3 \times 6$

The $2$nd Ulam number after $1$

The $2$nd positive integer after $1$ such that all smaller positive integers coprime to it are prime

The $2$nd generalized pentagonal number after $1$:
$2 = \dfrac {1 \paren {3 \times 1 + 1} } 2$

The $2$nd (strictly) positive integer after $1$ which cannot be expressed as the sum of exactly $5$ non-zero squares

The $2$nd highly composite number after $1$:
$\map \tau 2 = 2$

The $2$nd special highly composite number after $1$

The $2$nd highly abundant number after $1$:
$\map \sigma 2 = 3$

The $2$nd superabundant number after $1$:
$\dfrac {\map \sigma 2} 2 = \dfrac 3 2 = 1 \cdotp 5$

The $2$nd almost perfect number after $1$:
$\map \sigma 2 = 3 = 4 - 1$

The $2$nd factorial after $1$:
$2 = 2! = 2 \times 1$

The $2$nd superfactorial after $1$: