今パートは0.1～オイラーの定数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 |
|---|---|---|---|---|
| 0.1 | 10^(-1) | one tenth | 有 | ∞ |
| 0.108410151223111 | Trott constant T1 | |||
| 連分数展開で \(\displaystyle [0; 1, 0, 8, 4, 1, 0, 1, 5, 1, 2, 2, 3, 1, 1, 1, 3, 6, \ldots ] \) WMW、A039662 | ||||
| 0.107653919226484 | One-Ninth Constant | |||
| 逆数も定義されている (A073007) WMW、A072558 | ||||
| 0.110000999969134 | リウヴィル定数に近似する代数方程式の解 | Liouville’s constant | ||
| 10x6-75x3-190x+21=0 WMW、A093409 | ||||
| 0.110001 | リウヴィル定数 | Liouville’s constant | ||
| \(\displaystyle l = \sum_{n \geq 1} 10^{-n!} \) WMW、A012245 | ||||
| 0.114942044853296 | ケプラー=ブカンプ定数 | Kepler–Bouwkamp constant | ||
| \(\displaystyle \prod_{n \geq 3} \cos \frac{\pi}{n} \) 逆数もある WMW、A085365 | ||||
| 0.118419723766729 | Planck’s Radiation Function | |||
| \(\displaystyle f \left( x \right) = \frac{15}{\pi^{4} x^{5} \left( e^{x^{-1}} -1 \right)} \) の時、f”(x) = 0の解の一つ WMW、A133839 | ||||
| 0.123456789101112 | チャンパーノウン定数 | Champernowne constant | 超 | |
| \(\displaystyle C_{10} = \sum_{n \geq 1} \sum_{k=10^{n-1}}^{10^{n}-1} \frac{k}{10^{kn-9 \sum_{j=0}^{n-1} 10^{j} \left( n-j-1 \right)}} \\\displaystyle = \sum_{n \geq 1} \frac{n}{10^{n+ \sum_{k=1}^{n} \lfloor \log \left( k \right) \rfloor}} \) WMW、A033307 | ||||
| 0.147583617650433 | Plouffe’s Constant C | |||
| \(\displaystyle C = \frac{1}{\pi} \tan^{-1} \frac{1}{2} \\\displaystyle = \frac{1}{\pi} \sum_{n \geq 0} \frac{\left( -1 \right)^{n}}{\left( 2^{2n+1} \right) \left( 2n+1 \right)} \) WMW、A086203 | ||||
| 0.159154943091895 | Plouffe’s constant A | 超 | ||
| 1/2π WMW、A086201 | ||||
| 0.171500493141536 | Hall-Montgomery Constant | |||
| \(\displaystyle \delta_0 = 1 – \frac{\pi^2}{6} – \ln \left( 1 + \sqrt{e} \right) \ln \left( \frac{e}{(1 + \sqrt{e}} \right) + 2 \mathrm{Li}_2 \left( \frac{1}{1 + \sqrt{e}} \right) \\\displaystyle = 1 + \frac{\pi^2}{6} + 2 \mathrm{Li}_2 \left( – \sqrt{e} \right) \) WMW、A143301 | ||||
| 0.180550541849851 | Sinc Function | |||
| \(\displaystyle \prod_{n \geq 1} \mathrm{sinc} \frac{2 \pi}{2n+1} \) = π/2K WMW、A118253 | ||||
| 0.187859642462067 | MRB定数 | MRB constant | ||
| \(\displaystyle \sum_{k \geq 1} \left( -1 \right)^k \left( k^{k^{-1}} -1 \right) \) WMW、A037077 | ||||
| 0.194528049465325 | 2nd du Bois-Reymond constant | 超 | ||
| \(\displaystyle C_{2} = \frac{e^{2} – 7}{2} = \int_{0}^{\infty} \vert \frac{d}{dt} \left( \frac{\sin t}{t} \right)^{2} \vert dt -1 \) WMW | ||||
| 0.199458818 | Vallée Constant | 9? | ||
| WMW、A143302 | ||||
| 0.2 | Bruijn–Newman定数の上限値 | de Bruijn–Newman constant | ||
| 0.201405235272642 | Planck’s Radiation Function | |||
| \(\displaystyle f \left( x \right) = \frac{15}{\pi^{4} x^{5} \left( e^{x^{-1}} -1 \right)} \) の時、f'(x) = 0の解 WMW、A133838 | ||||
| 0.202456141492396 | White House Switchboard Constant | |||
| \(\displaystyle W = \exp \left( – \left( 1 + 8^{1/ \left( e-1 \right)} \right)^{1 / \pi} \right) \) 近似値を表す式が5種類ある WMW、A182064 | ||||
| 0.207879576350761 | i^iの主値 | 超 | ||
| \(\displaystyle e^{- \frac{\pi}{2}} \) WMW、A049006 | ||||
| 0.208987640249978 | リュカ数列の派生 | Lucas number | ||
| リュカ数列の10^n番目の桁数を小数展開した時の収束値 \(\displaystyle \log \phi = \lim_{n \to \infty} 10^{-n} \log \left( \left( \frac{1+ \sqrt{5}}{2} \right)^{10^n} + \left( \frac{1- \sqrt{5}}{2} \right)^{10^n} \right) \) ※φ：黄金比 WMW、A097348 | ||||
| 0.21862 | Cameron’s Sum-Free Set Constant | |||
| 上限値。下限値は0.21759 WMW | ||||
| 0.235711131719232 | コープランド・エルデシュ定数 | Copeland–Erdős constant | 無 | |
| \(\displaystyle C_{CE} = \sum_{n \geq 1} p_{n} 10^{-n- \sum_{k=1}^{n} \lfloor \log{p_{k}} \rfloor} \) \( p_n \)はn番目の素数 WMW、A033308 | ||||
| 0.2477 | 過剰数の自然密度 | |||
| ±0.0003の幅がある | ||||
| 0.261497212847642 | マイセル-メルテンス定数 | Meissel–Mertens constant | 超? | 8010 |
| \(\displaystyle M = B_{1} = \lim_{n \to \infty} \left( \sum_{p \leq n} \frac{1}{p} – \ln \left( \ln \left( n \right) \right) \right) \\\displaystyle = \gamma + \sum_{p} \left( \ln \left( 1- \frac{1}{p} \right) + \frac{1}{p} \right) \\\displaystyle = \gamma – \sum_{k \geq 1} \sum_{j \geq 2} \frac{1}{jp_{k}^{j}} = \gamma – \sum_{j \geq 2} \frac{P \left( n \right)}{n} \\\displaystyle = \gamma + \sum_{m \geq 2} \frac{\mu \left( m \right)}{m} \ln \left( \zeta \left( m \right) \right) \) γ：オイラーの定数、P(n)：素数ゼータ関数、μ(m)：メビウス関数 WMW、A077761 | ||||
| 0.270079723310951 | Plouffe’s Constants B’ | |||
| 1/π ⨁ 1/4π ⨁：2進数における排他的論理和 WMW、A111953 | ||||
| 0.273944195739271 | Trott Constants T2 | |||
| \(\displaystyle T_{2} = \frac{2}{7+ \frac{3}{9+ \frac{4}{4+ \cdots }}} \) WMW、A091694 | ||||
| 0.280169499023869 | バーシュタイン定数β | Bernstein’s constant | 超 | |
| WMW、A073001 | ||||
| 0.283757296380075 | Planck’s Radiation Function | |||
| \(\displaystyle f \left( x \right) = \frac{15}{\pi^{4} x^{5} \left( e^{x^{-1}} -1 \right)} \) の時、f”(x) = 0の解の一つ WMW、A133840 | ||||
| 0.284258224651343 | 掛谷問題 | Kakeya needle problem | ||
| ((5 – 2√2)π)/24 WMW、A093823 | ||||
| 0.286747428434478 | Strongly carefree constant | |||
| \(\displaystyle K_{2} = \prod_{p} \left( 1- \frac{3p-2}{p^3} \right) \\\displaystyle = \frac{6}{\pi^2}\prod_{p} \left( 1- \frac{1}{p^{2} +p} \right) \) WMW、A065473 | ||||
| 0.288788095086602 | Tree Searching | |||
| \(\displaystyle \prod_{n \geq 1} \left( 1-2^{-n} \right) \) WMW、A048651 | ||||
| 0.291560904030818 | Domino tiling | |||
| \(\displaystyle \frac{C}{\pi} = \int_{- \pi}^{\pi} \left( 4 \pi \right)^{-1} \cosh^{-1} \left( \sqrt{\frac{\cos t +3}{2}} \right) dt \) C：カタランの定数 WMW、A143233 | ||||
| 0.303663002898732 | ガウス=クズミン=ウィルズィング定数 | Gauss–Kuzmin–Wirsing constant | 385 | |
| WMW、A038517 | ||||
| 0.306348962530033 | Golden spiral | |||
| \(\displaystyle \vert b \vert = \frac{\ln \varphi}{\pi/2} \) φ：黄金比 Wiki、A212225 | ||||
| 0.306349 | Golden Rectangle b | |||
| \(\displaystyle b = 2 \pi^{-1} \ln \phi \) WMW | ||||
| 0.308443779561986 | Littlewood-Salem-Izumi Constant | |||
| WMW、A157957 | ||||
| 0.311078866704819 | Zolotarev-Schur Constant | |||
| \(\displaystyle \sigma = c^{-2} \left( 1 – \frac{E \left( c \right)}{K \left( c \right)} \right)^{2} \) →K(c)：第1種完全楕円積分、E(c)：第2種完全楕円積分、c：方程式の一意解 WMW、A143295 | ||||
| 0.312832929508881 | 積が奇数の素数をとる時のケプラー=ブカンプ定数 | Kepler–Bouwkamp constant | ||
| \(\displaystyle \prod_{p \geq 3} \cos \left( \frac{\pi}{p} \right) \) Wiki、A131671 | ||||
| 0.31830988618379 | 1/π | |||
| 関連記事 | ||||
| 0.322634098939244 | Squarefree | |||
| \(\displaystyle \prod_{p} \left( 1- \frac{2}{p^2} \right) \) = 2F-1 WMW、A065474 | ||||
| \( 0. \dot{3} \) | line line picking | |||
| line line pickingとは長さが1の線分の間に任意の2点を指定した時の新たな線分の長さの平均値を指す。line pickingには正方形や円などの派生種等多数存在する。特に立方体の時の定数をロビンス定数 (Δ(3)≒0.6617) と呼ぶ。 WMW | ||||
| 0.340537329550999 | Pólya Random walk constant | |||
| \(\displaystyle p \left( 3 \right) = 1- \left( \frac{3}{8 \pi^3} \int_{- \pi}^{\pi}\int_{- \pi}^{\pi}\int_{- \pi}^{\pi} \frac{dxdydz}{3-\cos{x} \cos{y} \cos{z}} \right)^{-1} \\\displaystyle = 1-16 \sqrt{\frac{2}{3}} \pi^{3} \left( \Gamma \left( \frac{1}{24} \right) \Gamma \left( \frac{5}{24} \right) \Gamma \left( \frac{7}{24} \right) \Gamma \left( \frac{11}{24} \right) \right)^{-1} \) WMW、A086230 | ||||
| 0.353236371854995 | ハフナー=サルナック=マッカーリー定数 | Hafner–Sarnak–McCurley constant | ||
| \(\displaystyle \lim_{n \to \infty} D(n) = \prod_{k \geq 1} \left\{ 1- \left[ 1- \prod_{j=1}^{n} \left( 1-p_{k}^{-j} \right) \right]^{2} \right\} \) WMW、A085849 | ||||
| 0.366512920581664 | ガンベル分布の中央値 | Median of the Gumbel distribution | ||
| \(\displaystyle ll_{2} = – \ln \left( \ln \left( 2 \right) \right) \) A074785 | ||||
| 0.367879441171442 | 秘書問題 | Sultan’s Dowry Problem | ||
| 1/e。応募者が十分に多い場合、最善の応募者を選択する確率。 WMW、A068985 | ||||
| 0.373955813619202 | アルティン定数 | Artin’s constant | ||
| \(\displaystyle C_{A} = \prod_{p} \left( 1- \frac{1}{p^{2} -p} \right) \) WMW、A005596 | ||||
| 0.412454033640107 | プロウエット=トゥエ=モールス定数 | Prouhet–Thue–Morse constant | 超 | |
| \(\displaystyle \sum_{n=0}^{\infty} \frac{t_{n}}{2^{n+1}} \\\displaystyle t_{0} = 0, \quad t_{2n+1} = 1-t_{n}, \quad t_{2n} = t_{n} \) WMW、A014571 | ||||
| 0.414682509851111 | 素数定数 | Prime constant | 無 | |
| \(\displaystyle \rho = \sum_{p} 2^{-p} \) WMW、A051006 | ||||
| 0.422157733115826 | 半径が1のルーローの四面体の体積 | Volume of Reuleaux tetrahedron | ||
| \(\displaystyle V_{R} = \frac{8 \pi}{3} – \frac{27}{4} \cos^{-1} \frac{1}{3} +\frac{\sqrt{2}}{4} \) WMW、A102888 | ||||
| 0.42513153135 | Aarex’s Funny Number | |||
| Pointless Gigantic List of Numbers | ||||
| 0.428249505677094 | Carefree constant | |||
| \(\displaystyle K_{1} = \prod_{p} \left( 1- \frac{2p-1}{p^3} \right) \) WMW、A065464 | ||||
| 0.438259147390354 | フレネル積分 | Fresnel integral | ||
| \(\displaystyle S \left( 1 \right) = \int_{0}^{1} \sin{\left( t^2 \right)} dt = \sum_{n=0}^{\infty} \frac{\left( -1 \right)^{n}}{\left( 2n+1 \right)! \left( 4n+3 \right)} \) 備考：\(\displaystyle S \left( x \right) = \int_{0}^{x} \sin{\left( t^2 \right)} dt = \sum_{n=0}^{\infty} \left( -1 \right)^{n} \frac{x^{4n+3}}{\left( 2n+1 \right)! \left( 4n+3 \right)} \) WMW | ||||
| 0.440053467052492 | Yff予想のABC-8ω^3の現在の最大値 | Yff Conjecture | ||
| この時の角度Aの値は約1.409356270 WMW、A133845 | ||||
| 0.471861653452681 | Bloch定数の上限値 | Bloch constant | ||
| \(\displaystyle B = \frac{1}{\sqrt{1 + \sqrt{3}}} \frac{\Gamma \left( 1/3 \right) \Gamma \left( 11/12 \right)}{\Gamma \left( 1/4 \right)} \\\displaystyle \sqrt{\pi} 2^{1/4} \frac{\Gamma \left( 1/3 \right)}{\Gamma \left( 1/4 \right)} \sqrt{\frac{\Gamma \left( 11/12 \right)}{\Gamma \left( 1/12 \right)}} \) 下限値は√3/4+2*10^(-4) (≒0.433212702) WMW、A085508 | ||||
| 0.47494937998792 | Weierstrass constant | |||
| \(\displaystyle \frac{2^{5/4} e^{\pi/8} \sqrt{\pi}}{\Gamma^{2} \left( 1/4 \right)} \) WMW、A094692 | ||||
| 0.475626076735988 | Plouffe’s Constants B | |||
| 1/π ⨁ 1/2π ⨁：2進数における排他的論理和 WMW、A086202 | ||||
| 0.482677281939181 | Trott Constants T3 | |||
| \(\displaystyle T_{3} =0+ \frac{4}{8+ \frac{2}{6+ \frac{7}{7+ \cdots }}} \) WMW、A113307 | ||||
| 0.494566817223496 | Shapiro’s Cyclic Sum Constant | |||
| WMW、A086277 | ||||
| 0.507833922868438 | ln(σ) | Somos’s Quadratic Recurrence Constant | ||
| \(\displaystyle \sum_{k \geq 1} 2^{-k} \ln k \) WMW、A114124 | ||||
| 0.523822571389864 | Chi(z)=0 | |||
| \(\displaystyle Chi \left( z \right) = \gamma + \ln z + \int_{0}^{z} \frac{\cosh t – 1}{t} dt = 0 \) WMW、A133746 | ||||
| 0.534949606142307 | Tree β | |||
| \(\displaystyle \beta = \frac{1}{\sqrt{2 \pi}} \left( 1 + \sum_{k \geq 2} T’ \left( \alpha^{-k} \right) \alpha^{-k} \right)^{3/2} \) WMW、A086308 | ||||
| 0.539645491190413 | Ioachimescu constant | |||
| \(\displaystyle 2+ \zeta \left( \frac{1}{2} \right) = 2- \left( 1+ \sqrt{2} \right) \sum_{n \geq 1} \frac{\left( -1 \right)^{n+1}}{\sqrt{n}} \) A242616 | ||||
| 0.543258965342976 | Bloch–Landau constant | |||
| \(\displaystyle L = \frac{\Gamma \left( \frac{1}{3} \right) \Gamma \left( \frac{5}{6} \right)}{\Gamma \left( \frac{1}{6} \right)} \\\displaystyle = \frac{\left( – \frac{2}{3} \right)! \left( – \frac{1}{6} \right)!}{\left( – \frac{5}{6} \right)!} \) 備考：Γ(n+1) = n! WMW、A081760 | ||||
| 0.553574358897045 | Golden Rhombus θ | |||
| \(\displaystyle \theta = \cot^{-1} \phi = 2^{-1} \tan^{-1} 2 \approx 31.7175^{\circ} \) WMW、A195693 | ||||
| 0.567143290409783 | オメガ定数 | Omega constant | 超 | |
| \(\displaystyle \Omega e^{\Omega} = 1 \\\displaystyle \Omega = W \left( 1 \right) = \sum_{n \geq 1} \frac{\left( -n \right)^{n-1}}{n!} \\\displaystyle = \lim_{n \to \infty} {}^{n} \left( e^{-1} \right) = e^{- \Omega} \) WMW、A030178 | ||||
| 0.567148130202017 | Andrica’s Conjecture | |||
| 127x-113x=1の解。2つの連続する素数p, qに対しqx-px=1となるような最小のx。 WMW、A038458 | ||||
| 0.56975158291971 | Weakly carefree constant | |||
| \(\displaystyle K_{3} = 2K_{1}-K_{2} \) K1…Carefree constant、K2…Strongly carefree constant WMW、A118261 | ||||
| 0.575959968892945 | スティーブンス定数 | Stephens’ constant | 超? | |
| \(\displaystyle C_{s} = \prod_{p} \left( 1- \frac{p}{p^{3}-1} \right) \) WMW、A065478 | ||||
| 0.577215664901532 | オイラーの定数 | Euler–Mascheroni constant | 無? | 477511832674 |
| \(\displaystyle \gamma = \lim_{n \to \infty} \left( – \ln{n} + \sum_{k=1}^{n} \frac{1}{k} \right) = \int_{1}^{\infty} \left( – \frac{1}{x} + \frac{1}{\lfloor x \rfloor} \right) dx \) スティルチェス定数γ0の値でもある WMW、A001620 | ||||
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