313

Number
$313$ (three hundred and thirteen) is:


 * The $65$th prime number


 * The $4$th prime number after $3$, $5$, $7$ which is palindromic in both decimal and binary:
 * $313_{10} = 100 \, 111 \, 001_2$


 * The lower end of the $6$th record-breaking gap between twin primes:
 * $347 - 313 = 34$


 * The $9$th integer after $0$, $1$, $3$, $5$, $7$, $9$, $33$, $99$ which is palindromic in both decimal and binary:
 * $313_{10} = 100 \, 111 \, 001_2$


 * The $9$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$, $53$, $73$:
 * $313$ is prime; $31$, $3$ are prime; $13$, $3$ are prime


 * The $11$th palindromic prime:
 * $2$, $3$, $5$, $7$, $11$, $101$, $131$, $151$, $181$, $191$, $313$, $\ldots$


 * The larger of the $20$th pair of twin primes, with $311$


 * The $24$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $181$, $193$, $223$, $229$, $233$, $257$, $263$, $269$


 * The $48$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$, $310$:
 * $313 \to 3^2 + 1^2 + 3^2 = 9 + 1 + 9 = 19 \to 1^2 + 9^2 = 1 + 81 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$

Also see