# 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