7

Number
$7$ (seven) is:


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


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


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


 * The start of an arithmetic progression of $6$ prime numbers:
 * $7, 37, 67, 97, 127, 157$


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


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


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


 * The smallest number that is not the sum of at most $3$ square numbers: see Integer as Sum of Three Squares


 * 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 $4$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 $2$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 $1$st prime number whose period is of maximum length:
 * $\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$


 * The $4$th of the sequence of $n$ such that $p_n \# + 1$, where $p_n \#$ denotes primorial of $n$, is prime, after $2, 3, 5$:
 * $p_7 \# + 1 = 2 \times 3 \times 5 \times 7 + 1 = 211$


 * The $2$nd of $5$ primes of the form $2 x^2 + 5$:
 * $2 \times 1^2 + 5 = 7$

Also see

 * Brocard's Problem