埃尔德什-波温常数

埃尔德什-波温常数
識別
種類	無理數
符號	$E$
位數數列編號	A065442
性質
定義	$E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}$
表示方式
值	1.60669515 $E=\sum _{n=1}^{\infty }{\frac {1}{2^{n^{2}}}}{\frac {2^{n}+1}{2^{n}-1}}$ $E=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}$ $E=1+\sum _{n=1}^{\infty }{\frac {1}{2^{n}(2^{n}-1)}}$ $E=\sum _{n=1}^{\infty }{\frac {\sigma _{0}(n)}{2^{n}}}$

二进制	1.100110110101000001011111…
八进制	1.466501374744176211033763…
十进制	1.606695152415291763783301…
十六进制	1.9B505F9E43F22437F372C528…

查论编

埃尔德什-波温常数是所有梅森数的倒数之和。

根据定义，它是：

E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}\approx 1.606695152415291763\dots

也可以写成以下的形式：

E=\sum _{n=1}^{\infty }{\frac {1}{2^{n^{2}}}}{\frac {2^{n}+1}{2^{n}-1}}

E=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}

E=1+\sum _{n=1}^{\infty }{\frac {1}{2^{n}(2^{n}-1)}}

E=\sum _{n=1}^{\infty }{\frac {\sigma _{0}(n)}{2^{n}}}

其中σ₀(n) = d(n)是因子函数，它是一个积性函数，是n的正因子的数目。

埃尔德什在1948年证明了E是一个无理数。

外部链接

埃里克·韦斯坦因. 埃尔德什-波温常数. MathWorld.

查论编無理數
柴廷常數 ( $Ω$ ) 刘维尔数質常數（英语：Prime constant） ( $ρ$ ) 欧米加常数卡漢常數 2的自然对数高斯常數 ( $G$ ) 2的12次方根阿培里常数 ( $ζ (3)$ ) 塑膠數 ( $ρ$ ) 2的算術平方根超黃金比例 ( $ψ$ ) 埃尔德什-波温常数 ( $E$ ) 黄金分割率 ( $φ$ ) 3的算術平方根 5的算術平方根白銀比例 ( $δ S$ ) 6的算術平方根（英语：Square root of 6） 7的算術平方根（英语：Square root of 7）自然常數/尤拉數 ( $e$ ) 圓周率 ( $π$ ) 青銅比例
分裂數（英语：Schizophrenic number）超越數三角函數數

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