2

Number
$2$ (two) is:


 * The successor integer to the number 1|$1$ (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 prime 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 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 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 positive integer 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:
 * $2 = \map {\operatorname {Ta} } 1 = 1^3 + 1^3$

$2$nd Term

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


 * 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 $1$:
 * $2 = 2\$ = 2! \times 1!$


 * The $2$nd factorion base $10$ after $1$:
 * $2 = 2!$


 * The $2$nd of the trivial $1$-digit pluperfect digital invariants after $1$:
 * $2^1 = 2$


 * The $2$nd of the (trivial $1$-digit) Zuckerman numbers after $1$:
 * $2 = 1 \times 2$


 * The $2$nd of the $5$ known powers of $2$ whose digits are also all powers of $2$:
 * $1$, $2$, $\ldots$


 * The $2$nd after $1$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes


 * The $2$nd of the (trivial $1$-digit) harshad numbers after $1$:
 * $2 = 1 \times 2$


 * The $2$nd Bell number after $(1)$, $1$


 * The $2$nd positive integer after $1$ whose cube is palindromic (in this case trivially):
 * $2^3 = 8$


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

$3$rd Term

 * The $3$rd integer $m$ after $0$, $1$ such that $m! + 1$ (its factorial plus $1$) is prime:
 * $2! + 1 = 2 + 1 = 3$


 * The $3$rd integer after $0$, $1$ such that its double factorial plus $1$ is prime:
 * $2!! + 1 = 3$


 * The $3$rd integer after $0$, $1$ which is (trivially) the sum of the increasing powers of its digits taken in order:
 * $2^1 = 2$


 * The $3$rd subfactorial after $0$, $1$:
 * $2 = 3! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} }$


 * The $3$rd integer $n$ after $-1$, $0$ such that $\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$ for integer $m$:
 * $\dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3 = 2^2$


 * The $3$rd integer $m$ after $0$, $1$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
 * $2^2 = \dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3$


 * The $3$rd palindromic integer after $0$, $1$ which is the index of a palindromic triangular number
 * $T_2 = 3$


 * The $3$rd integer $n$ after $0$, $1$ such that $2^n$ contains no zero in its decimal representation:
 * $2^2 = 4$


 * The $3$rd integer $n$ after $0$, $1$ such that $5^n$ contains no zero in its decimal representation:
 * $5^2 = 25$


 * The $3$rd integer $n$ after $0$, $1$ such that both $2^n$ and $5^n$ have no zeroes:
 * $2^2 = 4, 5^2 = 25$


 * The $3$rd integer after $0$, $1$ which is palindromic in both decimal and ternary:
 * $2_{10} = 2_3$


 * The $3$rd Fibonacci number after $1$, $1$:
 * $2 = 1 + 1$

Miscellaneous

 * The number of pairs of twin primes less than $10$:
 * $\tuple {3, 5}$, $\tuple {5, 7}$

Next in sequence: $3$




Next in sequence: $4$




Next in sequence: $5$




Next in sequence: $6$




Next in sequence: $9$ and above