4

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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 $n \nmid \paren {n - 1}!$

$1$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:
$4 = 2 + 2$

$2$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 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 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 $1$, $2$:
$\map \tau 4 = 3$

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 $1$, $2$, $3$:
$4 = 1 \times 4$

$5$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

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

There exist exactly $4$ Kepler-Poinsot polyhedra

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

Historical Note

The number $4$, as was $8$, was associated by the Pythagoreans with the concept of justice, being evenly balanced: $4 = 2 + 2$, where $2$ is the principle of diversity.

Throughout history, the number $4$ has been regarded with particular significance.

There were originally believed to be $4$ elements out of which everything was formed:

Earth, Air, Fire and Water.

There are $4$ humours:

Sanguine, Melacholic, Choleric and Phlegmatic.

There are $4$ cardinal points of the compass:

North, East, South and West.

There are $4$ seasons of the year:

Spring, Summer, Autumn and Winter.

In the Old Testament, there were $4$ rivers which watered the Garden of Eden:

Pishon, Gihon, the Tigris, and the Euphrates.

In Islam, these rivers are:

Saihan (Syr Darya), Jaihan (Amu Darya), Furat (Euphrates) and Nil (Nile).

In the New Testament, there are $4$ Gospels:

Matthew, Mark, Luke and John.

The most aesthetically pleasing musical intervals are those whose frequencies are associated with the ratio of $1 : 4$.