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 $2$, $3$, $5$
- The $4$th integer $n$ after $3$, $4$, $5$ for which the Ramanujan-Nagell equation $x^2 + 7 = 2^n$ has an integral solution:
- $11^2 + 7 = 128 = 2^7$
$5$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
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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