31

Number
$31$ (thirty-one) is:


 * The $11$th prime number, after $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$


 * The $2$nd of the $5$th pair of twin primes, with $29$


 * The $3$rd Mersenne number and the $3$rd Mersenne prime after $3$ and $7$:
 * $31 = 2^5 - 1$


 * The $9$th lucky number:
 * $1, 3, 7, 9, 13, 15, 21, 25, 31, \ldots$


 * The $3$rd emirp after $13, 17$.


 * The $8$th permutable prime after $2, 3, 5, 7, 11, 13, 17$.


 * The $8$th happy number after $1, 7, 10, 13, 19, 23, 28$:
 * $31 \to 3^2 + 1^2 = 9 + 1 = 10 \to 1^2 + 0^2 = 1$


 * The $6$th of the sequence of $n$ such that $p_n \# + 1$, where $p_n \#$ denotes primorial of $n$, is prime:
 * $2, 3, 5, 7, 11, 31$


 * The $2$nd of $29$ primes of the form $2 x^2 + 29$:
 * $2 \times 1^2 + 29 = 31$


 * The $21$st integer $n$ such that $2^n$ contains no zero in its decimal representation:
 * $2^{31} = 2 \, 147 \, 483 \, 648$


 * The $16$th positive integer which is not the sum of $1$ or more distinct squares:
 * $2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, \ldots$


 * Expressible as the sum of successive powers starting from $1$ in in $2$ different ways:
 * $31 = 1 + 5 + 5^2 = 1 + 2 + 2^2 + 2^3 + 2^4$

Also see