313

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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 $15$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$, $229$, $233$, $277$, $311$

The $18$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$, $233$, $239$, $293$, $311$

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

The $23$rd left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$, $167$, $173$, $197$, $223$, $283$

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 $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$, $310$:

$313 \to 3^2 + 1^2 + 3^2 = 9 + 1 + 9 = 19 \to 1^2 + 9^2 = 1 + 81 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$