7

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Number

$7$ (seven) is:

The $4$th prime number after $2$, $3$, $5$

$1$st Term

The $1$st prime number of the form $6 n + 1$:
$7 = 6 \times 1 + 1$

The $1$st long period prime:
$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$

The $1$st power of $7$ after the zeroth $1$:
$7 = 7^1$

The $1$st element of an arithmetic sequence of $6$ prime numbers:
$7$, $37$, $67$, $97$, $127$, $157$

The $1$st element of the $1$st pair of consecutive prime numbers different by $4$

The $1$st integer that is not the sum of at most $3$ square numbers

The $1$st integer the decimal representation of whose square can be split into two parts which are each themselves square:
$7^2 = 49$; $4 = 2^2$, $9 = 3^2$

$2$nd Term

The $2$nd safe prime after $5$:
$7 = 2 \times 3 + 1$

The $2$nd heptagonal number after $1$:
$7 = 1 + 7 = \dfrac {2 \paren {5 \times 2 - 3} } 2$

The $2$nd centered hexagonal number after $1$:
$7 = 1 + 6 = 2^3 - 1^3$

The $2$nd hexagonal pyramidal number after $1$:
$7 = 1 + 6$

The $2$nd second pentagonal number after $2$:
$7 = \dfrac {2 \paren {3 \times 2 + 1} } 2$

The $2$nd Mersenne number and Mersenne prime after $3$, leading to the $2$nd perfect number $28$:
$7 = 2^3 - 1$

The larger element of the $2$nd pair of twin primes, with $5$

The lower end of the $2$nd record-breaking gap between twin primes:
$11 - 7 = 4$

The $2$nd Woodall number after $1$, and $1$st Woodall prime:
$7 = 2 \times 2^2 - 1$

The $2$nd happy number after $1$:
$7 \to 7^2 = 49 \to 4^2 + 9^2 = 16 + 81 = 97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$

The $2$nd of $5$ primes of the form $2 x^2 + 5$:
$2 \times 1^2 + 5 = 7$ (Previous  ... Next)

The $2$nd Euclid number after $3$:
$7 = p_2\# + 1 = 2 \times 3 + 1$

The $2$nd positive integer $n$ after $4$ such that $n - 2^k$ is prime for all $k$

The $2$nd positive integer after $1$ whose divisor sum is a cube:
$\map {\sigma_1} 7 = 8 = 2^3$

$3$rd Term

The $3$rd positive integer after $1$, $2$ whose cube is palindromic:
$7^3 = 343$

The $3$rd Lucas prime after $2$, $3$

The $3$rd prime number after $2$, $3$ to be of the form $n! + 1$ for integer $n$:
$3! + 1 = 6 + 1 = 7$
where $n!$ denotes $n$ factorial

The $3$rd $n$ after $4$ and $5$, and the largest known, such that $n! + 1$ is square: see Brocard's Problem:
$7! + 1 = 5040 + 1 = 5041 = 71^2$

The $3$rd lucky number:
$1$, $3$, $7$, $\ldots$

The $3$rd palindromic lucky number:
$1$, $3$, $7$, $\ldots$

The $3$rd tri-automorphic number after $2$, $5$:
$7^2 \times 3 = 14 \mathbf 7$

The $3$rd Euclid prime after $2$, $3$:
$7 = p_2\# + 1 = 2 \times 3 + 1$

The $3$rd prime number after $3$, $5$ which is palindromic in both decimal and binary:
$7_{10} = 111_2$

$4$th Term

The $4$th (trivial, $1$-digit, after $2$, $3$, $5$) palindromic prime.

The $4$th permutable prime after $2$, $3$, $5$.

The $4$th generalized pentagonal number after $1$, $2$, $5$:
$7 = \dfrac {2 \paren {3 \times 2 + 1} } 2$

The index of the $4$th Mersenne prime after $2$, $3$, $5$:
$M_7 = 2^7 - 1 = 127$

The $4$th Lucas number after $(2)$, $1$, $3$, $4$:
$7 = 3 + 4$

The $4$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime, after $2$, $3$, $5$:
$7 \# + 1 = 2 \times 3 \times 5 \times 7 + 1 = 211$

The $4$th integer $n$ after $-1$, $0$, $2$ such that $\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$ for integer $m$:
$\dbinom 7 0 + \dbinom 7 1 + \dbinom 7 2 + \dbinom 7 3 = 8^2$

The $4$th positive integer after $2$, $3$, $4$ which cannot be expressed as the sum of distinct pentagonal numbers

The $4$th integer $m$ after $3$, $4$, $6$ such that $m! - 1$ (its factorial minus $1$) is prime:
$7! - 1 = 5040 - 1 = 5039$

The $4$th (trivially) two-sided prime after $2$, $3$, $5$

The $4$th prime number after $2$, $3$, $5$ consisting (trivially) of a string of consecutive ascending digits

The $4$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $\ldots$

The $4$th odd positive integer after $1$, $3$, $5$ such that all smaller odd integers greater than $1$ which are coprime to it are prime.

The $4$th odd positive integer after $1$, $3$, $5$ that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime

$5$th Term

The $5$th integer after $0$, $1$, $3$, $5$ which is palindromic in both decimal and binary:
$7_{10} = 111_2$

The $5$th integer $n$ after $3$, $4$, $5$, $6$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime:
$7! - 6! + 5! - 4! + 3! - 2! + 1! = 4421$

The index of the $5$th Mersenne number after $1$, $2$, $3$, $5$ which Marin Mersenne asserted to be prime

$6$th Term

The $6$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$ which cannot be expressed as the sum of exactly $5$ non-zero squares.

The $6$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

The smallest positive integer the decimal expansion of whose reciprocal has a period of $6$:
$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$

$7$th Term

The $7$th of the (trivial $1$-digit) pluperfect digital invariants after $1$, $2$, $3$, $4$, $5$, $6$:
$7^1 = 7$

The $7$th of the (trivial $1$-digit) Zuckerman numbers after $1$, $2$, $3$, $4$, $5$, $6$:
$7 = 1 \times 7$

The $7$th of the (trivial $1$-digit) harshad numbers after $1$, $2$, $3$, $4$, $5$, $6$:
$7 = 1 \times 7$

The $7$th after $1$, $2$, $3$, $4$, $5$, $6$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

$8$th Term

The $8$th number after $0$, $1$, $2$, $3$, $4$, $5$, $6$ which is (trivially) the sum of the increasing powers of its digits taken in order:
$7^1 = 7$

The $8$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$ such that $2^n$ contains no zero in its decimal representation:
$2^7 = 128$

The $8$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$ such that $5^n$ contains no zero in its decimal representation:
$5^7 = 78 \, 125$

The $8$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$ such that both $2^n$ and $5^n$ have no zeroes:
$2^7 = 128$, $5^7 = 78 \, 125$

Historical Note

The most obvious contemporary social significance of the number $7$ is the number of days in the week, which may have originated from the fact that it corresponds approximately with the number of days between specific phases of the moon.

Rational Diagonal

The number $7$ (seven) was referred to by the ancient Greeks as the rational diagonal of the $5 \times 5$ square.

This was on account of the fact that:

$5^2 + 5^2 = 50 \approx 7^2 = 49$

Linguistic Note

Words derived from or associated with the number $7$ include:

heptagon: a polygon with $7$ sides