4

Number
$4$ (four) is:


 * $2^2$


 * The only number which equals the number of letters in its name (four) when written in the English language


 * The only composite number $n$ such that $n \nmid \paren {n - 1}!$


 * The only number $n$ such that the alternating group $A_n$ is not simple.

$1$st Term

 * The $1$st semiprime:
 * $4 = 2 \times 2$


 * The $1$st power of $4$ after the zeroth $1$:
 * $4 = 4^1$


 * The $1$st Fermat pseudoprime to base $5$:
 * $5^4 \equiv 5 \pmod 4$


 * The $1$st $n$ such that $n! + 1$ is square: see Brocard's Problem


 * The $1$st positive integer $n$ such that $n - 2^k$ is prime for all $k$.


 * The $1$st (with $121$) of the $2$ square numbers which are $4$ less than a cube:
 * $4 = 2^2 = 2^3 - 4$


 * The $1$st in the sequence formed by adding the squares of the first $n$ primes:
 * $4 = \displaystyle \sum_{i \mathop = 1}^1 {p_i}^2 = 2^2$


 * The $1$st Smith number:
 * $4 = 2 + 2$


 * The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $1$ different way:
 * $4 = 1 + 3$


 * The number of primes with no more than $1$ digit:
 * $2$, $3$, $5$, $7$

$2$nd Term

 * The $2$nd square number, and the first square of a prime number:
 * $4 = 2 \times 2 = 2^2 = 1 + 3$


 * The $2$nd square number after $1$ to be the $\sigma$ (sigma) value of some (strictly) positive integer:
 * $4 = \map \sigma 3$


 * The $2$nd tetrahedral number after $1$:
 * $4 = 1 + 3 = \dfrac {2 \paren {2 + 1} \paren {2 + 2} } 6$


 * The $2$nd after $1$ of the $3$ tetrahedral numbers which are also square


 * The $2$nd trimorphic number after $1$:
 * $4^3 = 6 \mathbf 4$


 * The $2$nd power of $2$ after $(1)$, $2$:
 * $4 = 2^2$


 * The $2$nd powerful number after $1$


 * The $2$nd integer $m$ after $3$ such that $m! - 1$ (its factorial minus $1$) is prime:
 * $4! - 1 = 24 - 1 = 23$


 * The $2$nd integer $n$ after $3$ such that $m = \displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime:
 * $4! - 3! + 2! - 1! = 17$


 * The $2$nd even number after $2$ which cannot be expressed as the sum of $2$ composite odd numbers.


 * The $2$nd square after $1$ which has no more than $2$ distinct digits


 * The $2$nd even integer after $2$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes.


 * The $2$nd power of $2$ after $1$ which is the sum of distinct powers of $3$:
 * $4 = 2^2 = 3^0 + 3^1$


 * The $2$nd of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
 * $3$, $4$, $\ldots$

$3$rd Term

 * The $3$rd Lucas number after $(2)$, $1$, $3$:
 * $4 = 1 + 3$


 * The $3$rd highly composite number after $1$, $2$:
 * $\map {\sigma_0} 4 = 3$


 * The $3$rd superabundant number after $1$, $2$:
 * $\dfrac {\map \sigma 4} 4 = \dfrac 7 4 = 1 \cdotp 75$


 * The $3$rd almost perfect number after $1$, $2$:
 * $\map \sigma 4 = 7 = 8 - 1$


 * The $3$rd after $1$, $2$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.


 * The $3$rd of the $5$ known powers of $2$ whose digits are also all powers of $2$:
 * $1$, $2$, $4$, $\ldots$


 * The $3$rd positive integer after $2$, $3$ which cannot be expressed as the sum of distinct pentagonal numbers

$4$th Term

 * The $4$th Ulam number after $1$, $2$, $3$:
 * $4 = 1 + 3$


 * The $4$th highly abundant number after $1$, $2$, $3$:
 * $\map \sigma 4 = 7$


 * The $4$th (strictly) positive integer after $1$, $2$, $3$ which cannot be expressed as the sum of exactly $5$ non-zero squares.


 * The $4$th integer after $0$, $1$, $2$ which is palindromic in both decimal and ternary:
 * $4_{10} = 11_3$


 * The $4$th of the trivial $1$-digit pluperfect digital invariants after $1$, $2$, $3$:
 * $4^1 = 4$


 * The $4$th positive integer after $1$, $2$, $3$ such that all smaller positive integers coprime to it are prime


 * The $4$th of the (trivial $1$-digit) Zuckerman numbers after $1$, $2$, $3$:
 * $4 = 1 \times 4$


 * The $4$th of the (trivial $1$-digit) harshad numbers after $1$, $2$, $3$:
 * $4 = 1 \times 4$


 * The $4$th after $1$, $2$, $3$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

$5$th Term

 * The $5$th integer $n$ after $0$, $1$, $2$, $3$ such that $2^n$ contains no zero in its decimal representation:
 * $2^4 = 16$


 * The $5$th integer $n$ after $0$, $1$, $2$, $3$ such that $5^n$ contains no zero in its decimal representation:
 * $5^4 = 625$


 * The $5$th integer $n$ after $0$, $1$, $2$, $3$ such that both $2^n$ and $5^n$ have no zeroes:
 * $2^4 = 16$, $5^4 = 625$


 * The $5$th number after $0$, $1$, $2$, $3$ which is (trivially) the sum of the increasing powers of its digits taken in order:
 * $4^1 = 4$

Miscellaneous

 * The number of faces and vertices of a tetrahedron


 * The number of sides and vertices of a square


 * The number of faces which meet at each vertex of a regular octahedron


 * The number of dimensions in Einstein's space-time


 * There exist exactly $4$ Kepler-Poinsot polyhedra


 * Every positive integer can be expressed as the sum of at most $4$ squares.

Also see

 * Hyperbola can be Drawn through Four Non-Collinear Points
 * Plane Figure with Bilateral Symmetry about Two Lines has 4 Congruent Parts
 * Lagrange's Four Square Theorem, also represented as Hilbert-Waring Theorem for $2$nd Powers
 * Ferrari's Method
 * Four Color Theorem
 * Squares which are 4 Less than Cubes
 * Four Kepler-Poinsot Polyhedra
 * Brocard's Problem
 * Divisibility of n-1 Factorial by Composite n‎
 * Alternating Group is Simple except on 4 Letters

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