# 19

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## Number

$19$ (nineteen) is:

The $8$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$

The index of the $2$nd repunit prime after $R_2$:
$R_{19} = 1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$

The index of the $7$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$:
$M_{19} = 2^{19} - 1 = 524 \, 287$

The $2$nd of the $4$th pair of twin primes, with $17$

The lower end of the $3$rd record-breaking gap between twin primes:
$29 - 19 = 10$

The $3$rd centered hexagonal number after $1$, $7$:
$19 = 1 + 6 + 12 = 3^3 - 2^3$

The $2$nd Keith number after $14$:
$1$, $9$, $10$, $19$, $\ldots$

The $5$th happy number after $1$, $7$, $10$, $13$:
$19 \to 1^2 + 9^2 = 1 + 81 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$

The $3$rd prime number whose period is of maximum length:
$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$

The $3$rd integer after $7$, $13$ the decimal representation of whose square can be split into two parts which are each themselves square:
$19^2 = 361$; $36 = 6^2$, $1 = 1^2$

The $16$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$, $16$, $18$ such that $2^n$ contains no zero in its decimal representation:
$2^{19} = 524 \, 288$

Every positive integer can be expressed as the sum of at most $19$ $4$th powers.

The $10$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $\ldots$

The $12$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$ which cannot be expressed as the sum of distinct pentagonal numbers.

The $3$rd of $11$ primes of the form $2 x^2 + 11$:
$2 \times 2^2 + 11 = 19$ (Previous  ... Next)