7

From ProofWiki
Jump to navigation Jump to search

Previous  ... Next

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


The $1$st element of an arithmetic progression 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:
$7^2 = 49$; $4 = 2^2$, $9 = 3^2$


The $1$st prime number whose period is of maximum length:
$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$


$2$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 $2$nd 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 $1$ the sum of whose divisors is a cube:
$\map \sigma 7 = 8 = 2^3$


$3$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 $2$, $3$:
$7 = p_2\# + 1 = 2 \times 3 + 1$


$4$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


$5$th Term

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


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


$7$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 $1$, $2$, $3$, $4$, $5$, $6$:
$7 = 1 \times 7$


$8$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 in Sequence: $2$


Previous in Sequence: $3$


Previous in Sequence: $4$


Previous in Sequence: $5$


Previous in Sequence: $6$


Next in Sequence: $10$ and above


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:

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


Sources