数の比較Part05

2020.06.06
何のひねりもない数の比較の一覧表第5弾です。
今パートは√2～2までの数を扱います。
未掲載の数や数式も随時追加の予定です。

表の見方
種別欄：自…自然数、整…整数、有…有理数、無…無理数、代…代数的数、超…超越数
特に断りのない限り、数式中のlog(x)は常用対数、ln(x)は自然対数、pは素数であるものとします。


値 名称 英語名 種別 判明済桁数
1.41421356237309√2代10兆
WMW
1.44466786100976Steiner number, Iterated exponential Constant超
\(\displaystyle e^{e^{-1}} \)
\(\displaystyle \lim_{n \to \infty} {}^{n} a \) が収束するaの上限値 (この時の値はe)
WMW1、WMW2、A073229
1.45136923488338ラマヌジャン・ソルドナー定数 μRamanujan–Soldner constant無75500
\(\displaystyle \mathrm{li} \left( x \right) = \int_{0}^{x} \frac{dt}{\ln t} = 0 \)
WMW、A070769
1.45607494858268バックハウス定数Backhouse’s constant
\(\displaystyle \lim_{k \to \infty} \vert \frac{q_{k+1}}{q_{k}} \vert \\\displaystyle Q \left( x \right) = \frac{1}{P \left( x \right)} = \sum_{k \geq 1} q_{k} x^{k} \\\displaystyle P \left( x \right) = \sum_{k \geq 1} p_{k} x^{k} , \quad p_{k} \in \mathbb{ P } \)
WMW、A072508
1.46099848620631Baxter’s Four-coloring constant
\(\displaystyle \prod_{n \geq 1} \frac{\left( 3n-1 \right)^{2}}{3n \left( 3n-2 \right)} = \frac{3 \Gamma^3 \left( 1/3 \right)}{4 \pi^2} \\\displaystyle = \frac{1}{2^{1/3} L} = \frac{2 \pi}{\sqrt{3} \Gamma^3 \left( 2/3 \right)} \)
L：ランダウの定数
WMW、A224273
1.46557123187676超黄金比Supergolden ratio代
x3-x2-1=0の解の一つ。
\(\displaystyle \psi = \frac{1+ \sqrt[3]{\frac{29 + 3 \sqrt{93}}{2}} + \sqrt[3]{\frac{29 – 3 \sqrt{93}}{2}}}{3} \\\displaystyle \frac{2}{3} \cosh \left( \frac{\cosh^{-1} \left( 29/3 \right)}{3} \right) + \frac{1}{3} \)
Wiki、A092526
1.46707807943397ポーター定数Porter’s constant
\(\displaystyle C = \frac{6 \ln 2}{\pi^2} \left[ 3 \ln 2+4 \gamma – \frac{24}{\pi^2} \zeta’ \left( 2 \right) -2 \right] – \frac{1}{2} \\\displaystyle = \frac{6 \ln 2}{\pi^2} \left[ 3 \ln 2+4 \gamma + \frac{24}{\pi^2} \sum_{k \geq 2} k^{-2} \ln{k} -2 \right] – \frac{1}{2} \\\displaystyle = \frac{6 \ln 2}{\pi^2} \left[ 3 \ln 2+4 \gamma + \frac{24}{\pi^2} \frac{\pi^2}{6} \left( 12 \ln A – \gamma – \ln \left( 2 \pi \right) \right) -2 \right] – \frac{1}{2} \\\displaystyle = \frac{6 \ln 2 \left( 48 \ln A – \ln 2 -4 \ln \pi -2 \right)}{\pi^2} – \frac{1}{2} \)
γ：オイラーの定数、A：グレシャー・キンキリン定数
WMW、A086237
1.48203750177011五角数の逆数和Pentagonal number
\(\displaystyle \sum_{n \geq 1} \frac{2}{n \left( 3n-1 \right)} = 3 \ln 3 – 3^{-1} \pi \sqrt{3} \)
WMW、A244641
1.50304808247533Hard Square Entropy Constant
参考1、参考2
WMW、A085850
1.51638605915197Watson’s integral
WMW、A086231
1.5198177546350615/π^2
\(\displaystyle \prod_p \left( 1 + p^{-2} \right) = 15 \pi^{-2} \)
メビウス関数、素数和、素数積で現れる数の一つ。
A082020
1.52362708620249ドラゴン曲線Fractal dimension of the boundary of the dragon curve超
\(\displaystyle C_{d} = \log_{2} \left( \frac{1+ \sqrt[3]{73+ 6 \sqrt{87}} – \sqrt[3]{73- 6 \sqrt{87}}}{3} \right) \)
WMW、Wiki、A272031
1.52513527616098Continued Fraction Constants
\(\displaystyle 1+ K_{n \geq 1} \frac{n}{1} = 1 + \frac{1}{1+ \frac{2}{1+ \frac{3}{1+ \frac{4}{1+ \frac{5}{1+ \frac{6}{1+ \ddots}}}}}} \)
WMW、A111129
1.52995403705719omega_2 Constant
\(\displaystyle \omega_{2} = \frac{\Gamma^{3} \left( 1/3 \right)}{4 \pi} \)
WMW、A064582
1.539600717839リーブの四角氷定数Lieb’s square ice constant代
\(\displaystyle W_{2D} = \frac{8}{3 \sqrt{3}} \)
WMW、A118273
1.54149408253679Continued Fraction Constants
\(\displaystyle 1+ K_{n \geq 1} \frac{2n}{2n+1} = 1 + \frac{2}{3+ \frac{4}{5+ \frac{6}{7+ \frac{8}{9+ \frac{10}{11+ \frac{12}{13+ \ddots}}}}}} \\\displaystyle = \left( \sqrt{e} -1 \right)^{-1} \)
WMW、A113011
1.55138752454832Calabi triangle constant代
\(\displaystyle C_{CR} = 3^{-1} \cdot 2^{-2 / 3} \left( 2^{2 / 3} + \sqrt[3]{-23+3i \sqrt{237}} + \sqrt[3]{-23-3i \sqrt{237}} \right) \\\displaystyle = \frac{1}{3} \left( 1+ \sqrt{22} \cos \left( \frac{1}{3} \cos^{-1} \left( – \frac{23}{11 \sqrt{22}} \right) \right) \right) \)
2x3 – 2x2 -3x +2 = 0の解
WMW、A046095
1.559610469462362の超平方根Super square
xx=2
A030798
1.561552812808832の三角根Triangular root代
\(\displaystyle R_{2} = \frac{\sqrt{17} -1}{2} \\\displaystyle = \sqrt{4+ \sqrt{4+ \sqrt{4+ \sqrt{4+ \cdots}}}} -1 \\\displaystyle = \sqrt{4- \sqrt{4- \sqrt{4- \sqrt{4- \cdots}}}} \)
A222133
1.57079632679489ウォリス積Wallis product
\(\displaystyle \frac{\pi}{2} = \prod_{n \geq 1} \frac{4n^2}{4n^{2}-1} \\\displaystyle = \prod_{n \geq 1} \left( \frac{2n}{2n-1} \cdot \frac{2n}{2n+1} \right) \)
WMW
1.58496250072115Hausdorff dimension, Sierpinski triangle超
\(\displaystyle \log_{2} 3 \)
Stolarsky-Harborth定数にも関わる
WMW、A020857
1.606695152415エルデシュ-ボールウェイン定数Erdős–Borwein constant無
\(\displaystyle \sum_{m \geq 1} \sum_{n \geq 1} \frac{1}{2^{mn}} = \sum_{n \geq 1} \frac{1}{2^{n}-1} \)
探索木の定数αの値でもある。探索木に関連する定数が複数あり
WMW、A065442
1.61111492580837指数階乗定数Exponential factorial constant 超
\(\displaystyle S_{Ef} = \sum_{n \geq 1} \frac{1}{a_{n}} \\\displaystyle a_{0} = 1, a_{n} = n^{a_{n-1}} \)
WMW、A080219
1.61803398874989黄金比Golden ratio代
\(\displaystyle \phi = \frac{1+ \sqrt{5}}{2} \)
x2-x-1=0の解
WMW、A001622
1.64218843522212Lebesgue constant L2超
\(\displaystyle \frac{1}{5} + \frac{\sqrt{25 – 2 \sqrt{5}}}{\pi} \\\displaystyle = \frac{1}{\pi} \int_{0}^{\pi} \frac{| \sin \left( 5t/2 \right) |}{\sin \left( t/2 \right)} dt \)
WMW、A226655
1.64493406684822ζ(2) (リーマンゼータ関数)Riemann zeta function超
\(\displaystyle \zeta \left( 2 \right) = \sum_{k \geq 1} k^{-2} = \frac{\pi^2}{6} \)
WMW、A013661
1.66168794963359ソモス二次再帰定数Somos’ quadratic recurrence constant超?
\(\displaystyle \sigma = \prod_{n \geq 1} n^{2^{-n}} \)
WMW、A112302
1.70130161670407Golden Rhombus p
\(\displaystyle \frac{2}{\sqrt{1+ \phi^{-2}}} = \csc \frac{\pi}{5} \\\displaystyle = \sqrt{2+ \frac{2}{\sqrt{5}}} \)
WMW、A121570
1.70521114010536ニーヴン定数Niven’s constant
\(\displaystyle 1+ \sum_{n \geq 2} \left( 1- \frac{1}{\zeta \left( n \right)} \right) \)
WMW、A033150
1.71400629359161Smarandache Constants
WMW、A048834
1.73205080756887√3代
WMW
1.74540566240734ヒンチン調和平均Khinchin harmonic mean
\(\displaystyle K_{1} = \ln 2 \left( \sum_{n \geq 1} \ln \left( 1- \left( n+1 \right)^{2} \right)^{-1/n} \right)^{-1} \\\displaystyle = – \ln 2 \left( \sum_{n \geq 1} n^{-1} \ln \left( 1- \left( n+1 \right)^{2} \right) \right)^{-1} \\\displaystyle = n \left( \sum_{k \geq 1} a_{k}^{-1} \right)^{-1} \)
ak：連分数展開[a0; a1, a2, …]の要素
WMW、A087491
1.757932756618Kasner number
\(\displaystyle R = \sqrt{1+ \sqrt{2+ \sqrt{3+ \sqrt{4+ \cdots}}}} \)
WMW、A072449
1.75874362795118Infinite product constant
\(\displaystyle Pr_{1} = \prod_{n \geq 2} \left( 1+ \frac{1}{n} \right)^{1/n} \)
A242623
1.77245385090551ガウス積分Gaussian integral, Carlson–Levin constant 超
\(\displaystyle \sqrt{\pi} = \int_{- \infty}^{\infty} e^{-x^2} dx \\\displaystyle \left( – \frac{1}{2} \right) = \int_{0}^{1} \frac{dx}{\sqrt{- \ln x}} \)
A002161
1.78107241799019Exp.gamma, Barnes G-function, Mertens Theorem
\(\displaystyle e^{\gamma} = \prod_{n \geq 1} \frac{e^{n^{-1}}}{1+ n^{-1}} \\\displaystyle = \prod_{n \geq 1} \left( \prod_{k=0}^{n} \left( k+1 \right)^{\left( -1 \right)^{k+1} {n \choose k}} \right)^{\frac{1}{n+1}} \\\displaystyle = \lim_{n \to \infty} \frac{1}{\ln p_{n}} \prod_{k=1}^{n} \frac{1}{1-p_{k}^{-1}} \)
γ：オイラーの定数
WMW、A073004
1.78221397819136Grothendieck’s Constant kR
\(\displaystyle k_{R} = \frac{\pi}{2 \ln \left( 1+ \sqrt{2} \right)} \)
WMW、A088367
1.78657645936592Silverman constant
\(\displaystyle \prod_{p} \left( 1+ \sum_{k \geq 1} \frac{1}{p^{2k} – p^{k-1}} \right) \\\displaystyle = \sum_{n \geq 1} \frac{1}{\phi \left( n \right) \sigma_{1} \left( n \right)} \)
→φ(n), σ1(n)
WMW、A093827
1.78723165018296Komornik–Loreti constant q超
\(\displaystyle 1 = \sum_{k \geq 1} t_k q^{-k} \\\displaystyle \prod_{k \geq 0} \left( 1-q^{-2^k} \right) +\frac{q-2}{q-1} = 0 \)
WMW、A055060
1.82282524967884Masser-Gramain Constant
1+4c/π
WMW、A086058
1.83928675521416トリボナッチ定数Tribonacci Constant代
x3-x2-x-1=0の解の一つ
x+x-3=2
\(\displaystyle \phi_{3} = \frac{1+ \sqrt[3]{19 + 3 \sqrt{33}} + \sqrt[3]{19 – 3 \sqrt{33}}}{3} \\\displaystyle = 1+ \left( \sqrt[3]{\frac{1}{2} + \sqrt[3]{\frac{1}{2} + \sqrt[3]{\frac{1}{2} + \cdots}}} \right)^{-1} \)
WMW、A058265
1.84775906502257Connective constant代
\(\displaystyle \mu = \sqrt{2+ \sqrt{2}} \)
x4-4x2+2=0の解の一つ
A179260
1.85193705198246ギブズ定数Gibbs constant
\(\displaystyle G’ = Si \left( \pi \right) = \int_{0}^{\pi} \frac{\sin t}{t} dt \sum_{n \geq 1} \frac{\left( -1 \right)^{n-1} \pi^{2n-1}}{\left( 2n-1 \right) \left( 2n-1 \right)!} \\\displaystyle \int_{0}^{\pi} \mathrm{sinc} \theta d \theta \)
WMW、A036792
1.85326844870798ラプラス限界でレンズ状に囲まれた領域の面積Laplace Limit
WMW、A140133
1.85407467730137Gauss’ Lemniscate constant
\(\displaystyle \omega = \frac{L}{2 \sqrt{2}} = \int_{0}^{\infty} \frac{dx}{\sqrt{1+x^4}} \\\displaystyle = \frac{\Gamma^2 \left( 1/4 \right)}{4 \sqrt{\pi}} \)
WMW、A093341
1.86002507922119Spiral of Theodorus
\(\displaystyle T = \sum_{k \geq 1} \frac{1}{\sqrt{k^3} + \sqrt{k}} \\\displaystyle = 1/2 – \sum_{k \geq 1} \left( -1 \right)^k \left[ \zeta \left( k+1/2 \right) -1 \right] \)
WMW、A226317
1.89511781635593Ei(1)
\(\displaystyle – \int_{-1}^{\infty} \frac{e^{-t}}{t} dt = \gamma + \ln 1 + \sum_{n \geq 1} \frac{1}{n \cdot n!} \)
備考： \(\displaystyle \gamma + \ln z + \sum_{n \geq 1} \frac{z^n}{n \cdot n!} \)
WMW、A091725
1.902160583104双子素数におけるブルン定数Brun’s constant (Brun’s theorem)12
\(\displaystyle \sum_{p, p+2 \in \mathbb{ P }} \left( \frac{1}{p} + \frac{1}{p+2} \right) \)
WMW、A065421
1.92756197548292テトラナッチ定数Tetranacci constant 代
x4-x3-x2-x-1=0の解の一つ
x+x-4=2
WMW、A086088
1.92878218715021prime-generating constant
\(\displaystyle \lfloor \left( 2 \uparrow \right)^{n} \omega \rfloor = p \\\displaystyle \lfloor 2^{\omega} \rfloor = 3, \lfloor 2^{2^{\omega}} \rfloor = 13, \lfloor 2^{2^{2^{\omega}}} \rfloor = 16381, \cdots \)
WMW、A086238
1.94359643682075ランダウトーシェント定数Landau’s totient constant
\(\displaystyle \prod_{p} \left( 1+ \frac{1}{p^2 – p} \right) \\\displaystyle = \frac{\zeta \left( 2 \right) \zeta \left( 3 \right)}{\zeta \left( 6 \right)} = \frac{315 \zeta \left( 3 \right)}{2 \pi^4} \)
WMW1、WMW2、A082695
1.954085357876トーシェント関数の関連する値Totient Function
\(\displaystyle \sum_{n \geq 1} \frac{\phi \left( n \right)}{n!} \)
A000010
1.96285817320964Reciprocal Lucas Constant
\(\displaystyle P_{L} = \sum_{n \geq 1} L_{n}^{-1} \)
WMW、A093540
1.96594823664548Pentanacci Constant代
x5-x4-x3-x2-x-1=0の解の一つ
WMW、A103814
1.98358284342432Hexanacci Constant代
x6-x5-x4-x3-x2-x-1=0の解の一つ
WMW、A118427
1.99196419660503Heptanacci Constant代
x7-x6-x5-x4-x3-x2-x-1=0の解の一つ
WMW、A118428
2三角数の逆数和自∞

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