# 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 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) left-truncatable prime after $2$, $3$, $5$

- The $4$th (trivially) right-truncatable prime after $2$, $3$, $5$

- 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

- The $4$th minimal prime base $10$ after $2$, $3$, $5$

### $5$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

- The number of integer partitions for $5$:
- $\map p 5 = 7$

### $6$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:
- $2^7 = 128$, $5^7 = 78 \, 125$

## Also see

### Previous in Sequence: $1$

*Previous ... Next*: Happy Number*Previous ... Next*: Heptagonal Number*Previous ... Next*: Centered Hexagonal Number*Previous ... Next*: Hexagonal Pyramidal Number*Previous ... Next*: Woodall Number*Previous ... Next*: Sequence of Powers of 7*Previous ... Next*: Integers whose Divisor Sum is Cube

### Previous in Sequence: $2$

*Previous ... Next*: Sequence of Integers whose Cube is Palindromic*Previous ... Next*: Second Pentagonal Number*Previous ... Next*: Sum of 4 Consecutive Binomial Coefficients forming Square

### Previous in Sequence: $3$

*Previous ... Next*: Lucky Number*Previous ... Next*: Sequence of Palindromic Lucky Numbers*Previous ... Next*: Lucas Prime*Previous ... Next*: Mersenne Number*Previous ... Next*: Euclid Number*Previous ... Next*: Euclid Prime*Previous ... Next*: Mersenne Prime*Previous ... Next*: Prime Numbers of form Factorial Plus 1

### Previous in Sequence: $4$

*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Lucas Number*Previous ... Next*: Integers such that Difference with Power of 2 is always Prime

### Previous in Sequence: $5$

*Previous ... Next*: Palindromes in Base 10 and Base 2*Previous ... Next*: Odd Integers whose Smaller Odd Coprimes are Prime*Previous ... Next*: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares

*Previous ... Next*: Prime Number*Previous ... Next*: Safe Prime*Previous ... Next*: Palindromic Prime*Previous ... Next*: Twin Primes*Previous ... Next*: Permutable Prime*Previous ... Next*: Record Gaps between Twin Primes*Previous ... Next*: Sequence of Prime Primorial plus 1*Previous ... Next*: Integer Partition*Previous ... Next*: Minimal Prime

*Previous ... Next*: Generalized Pentagonal Number*Previous ... Next*: Index of Mersenne Prime*Previous ... Next*: Mersenne Prime/Historical Note*Previous ... Next*: Left-Truncatable Prime*Previous ... Next*: Right-Truncatable Prime*Previous ... Next*: Two-Sided Prime*Previous ... Next*: Prime Numbers Composed of Strings of Consecutive Ascending Digits*Previous ... Next*: Tri-Automorphic Number*Previous ... Next*: Palindromic Primes in Base 10 and Base 2

### Previous in Sequence: $6$

*Previous ... Next*: Pluperfect Digital Invariant*Previous ... Next*: Zuckerman Number*Previous ... Next*: Harshad Number*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Numbers not Sum of Distinct Squares*Previous ... Next*: Numbers which are Sum of Increasing Powers of Digits*Previous ... Next*: Sum of Sequence of Alternating Positive and Negative Factorials being Prime*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1

*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation*Previous ... Next*: Powers of 2 and 5 without Zeroes*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares

### Previous in Sequence: Above $6$

### Next in Sequence: $10$ and above

*Next*: Squares whose Digits can be Separated into 2 other Squares*Next*: Long Period Prime*Next*: Woodall Prime

## 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:

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $7$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $7$

Categories:

- Happy Numbers/Examples
- Heptagonal Numbers/Examples
- Centered Hexagonal Numbers/Examples
- Pyramidal Numbers/Examples
- Woodall Numbers/Examples
- Powers of 7/Examples
- Second Pentagonal Numbers/Examples
- Lucky Numbers/Examples
- Lucas Primes/Examples
- Mersenne Numbers/Examples
- Euclid Numbers/Examples
- Euclid Primes/Examples
- Mersenne Primes/Examples
- Lucas Numbers/Examples
- Prime Numbers/Examples
- Safe Primes/Examples
- Palindromic Primes/Examples
- Twin Primes/Examples
- Permutable Primes/Examples
- Integer Partitions/Examples
- Minimal Primes/Examples
- Generalized Pentagonal Numbers/Examples
- Indices of Mersenne Primes/Examples
- Mersenne's Assertion/Examples
- Left-Truncatable Primes/Examples
- Right-Truncatable Primes/Examples
- Two-Sided Primes/Examples
- Tri-Automorphic Numbers/Examples
- Pluperfect Digital Invariants/Examples
- Zuckerman Numbers/Examples
- Harshad Numbers/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Long Period Primes/Examples
- Woodall Primes/Examples
- Specific Numbers
- 7