157

Number
$157$ (one hundred and fifty-seven) is:


 * The $37$th prime number


 * The $1$st positive integer $n$ for which $\map \phi {2 n + 1} < \map \phi {2 n}$:
 * $\map \phi {2 \times 157 + 1} = 144$, $\map \phi {2 \times 157} = 156$


 * The $3$rd balanced prime after $5$, $53$:
 * $157 = \dfrac {151 + 163} 2$


 * The $8$th prime number after $53$, $71$, $103$, $107$, $109$, $149$, $151$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$


 * The $9$th of $29$ primes of the form $2 x^2 + 29$:
 * $2 \times 8^2 + 29 = 157$


 * The $12$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$


 * The $39$th and largest odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
 * $1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $89$, $97$, $101$, $103$, $115$, $157$

Also see