# 157

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## 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 the difference between a power of $2$ and a power of $3$.

- 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

*Previous ... Next*: Balanced Prime

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $157$