# 37

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## Number

$37$ (thirty-seven) is:

The $12$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$

The $4$th centered hexagonal number after $1$, $7$, $19$:
$37 = 1 + 6 + 12 + 18 = 4^3 - 3^3$

The $11$th lucky number:
$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $\ldots$

The $4$th emirp after $13$, $17$, $31$

The $9$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$

The $2$nd integer which has a reciprocal whose period is $3$:
$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

The $27$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
$2^{37} = 137 \, 438 \, 953 \, 472$

The $23$rd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$, $33$ which cannot be expressed as the sum of distinct pentagonal numbers

The $4$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$:
$37 = 6^2 + 1$

Every positive integer can be expressed as the sum of at most $37$ positive $5$th powers

The $7$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
$0$, $1$, $2$, $3$, $11$, $27$, $37$

The $4$th prime $p$ after $11$, $23$, $29$ such that the Mersenne number $2^p - 1$ is composite

The $6$th two-sided prime after $2$, $3$, $5$, $7$, $23$:
$37$, $3$, $7$ are prime

The $5$th of $5$ primes of the form $2 x^2 + 5$:
$2 \times 4^2 + 5 = 37$ (Previous)

The $3$rd of $29$ primes of the form $2 x^2 + 29$:
$2 \times 2^2 + 29 = 37$ (Previous  ... Next)