今パートはオイラーの定数~1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 |
|---|---|---|---|---|
| 0.577215664901532 | 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 | ||||
| 0.57786367489546 | Masser-Gramain Constant | |||
| π/2e WMW、A086056 | ||||
| 0.580577558204892 | Pell constant | 超? | ||
| \(\displaystyle \mathcal{P}_{P} = 1- \prod_{n \geq 0} \left( 1- 2^{-2n-1} \right) \) WMW、A141848 | ||||
| 0.581932705608592 | inflection point of x^(1/x) | |||
| WMW、A093157 | ||||
| 0.58194865931729 | Landau-Ramanujan Constant C | |||
| \(\displaystyle C = \frac{1}{2} \left( 1- \ln \frac{\pi e^{\gamma}}{2L} \right) – \frac{1}{4} \frac{d}{ds} \left( \ln \prod_{p \equiv 3 \pmod 4} \frac{1}{1- p^{-2s}} \right)_{s=1} \) →γ、L WMW、A227158 | ||||
| 0.581976706869326 | Continued Fraction Constants | |||
| \(\displaystyle C_{2} = K_{n \geq 1} \frac{n}{n} = \frac{1}{1+ \frac{2}{2+ \frac{3}{3+ \frac{4}{4+ \frac{5}{5+ \frac{6}{6+ \ddots}}}}}} \\\displaystyle = \left( \frac{\Gamma \left( n+2 \right)}{! \left( n+1 \right)} – 1 \right)^{-1} \\\displaystyle = \frac{1}{e-1} \) WMW、A073333 | ||||
| 0.592632718201636 | Lehmer’s Constant | |||
| 関連 WMW、A030125 | ||||
| 0.596347362323194 | オイラー-ゴンペルツ定数 | Euler–Gompertz constant | 無 | |
| \(\displaystyle G = \int_{0}^{\infty} \frac{e^{-x}}{1+x} dx = \int_{0}^{1} \frac{1}{1- \ln x} dx \\\displaystyle = \int_{0}^{\infty} \ln \left( 1+x \right) e^{-x} dx \\\displaystyle = \frac{1}{1+ \frac{1}{1+ \frac{1}{1+ \frac{2}{1+ \frac{2}{1+ \frac{3}{1+ \frac{3}{1+ \frac{4}{1+ \ddots}}}}}}}} \\\displaystyle = \frac{1}{2- \frac{1^2}{4- \frac{2^2}{6- \frac{3^2}{8- \frac{4^2}{10- \frac{5^2}{12- \frac{6^2}{14- \frac{7^2}{16- \ddots}}}}}}}} \) -eEi(-1) WMW、A073003 | ||||
| 0.598958167538433 | 3進数におけるチャンパーノウン定数 | Ternary Champernowne constant | ||
| C3=0.(1)(2)(10)(11)(12)(20)…3 WMW、A077771 | ||||
| 0.599070117367796 | 2nd lemniscate constant | |||
| L2 = 1/2G WMW、A076390 | ||||
| 0.602059991327962 | カタラン数が関連する数 | Catalan Number | ||
| カタラン数列の10^n番目の桁数を小数展開した時の収束値 \(\displaystyle \log 4 = \lim_{n \to \infty} \log \frac{C_{10^n}}{10^n} \\\displaystyle = \lim_{n \to \infty} \log \frac{\left( 2n \right)!}{10^{n} n! \left( n+1 \right)!} \) 中心二項係数の10^n番目の桁数を小数展開した時の収束値でもある \(\displaystyle \log 4 = \lim_{n \to \infty} \frac{\lceil \log {2 \cdot 10^{n} \choose 10^{n}} \rceil}{10^{n}} \) その他、中央二項係数に関する多数の式・値あり WMW、A114493 | ||||
| 0.605443657196732 | QRS Constant | |||
| \(\displaystyle S \left( N, a \right) = \sum_{i=1}^{N} \left( 1 – a^{2} \left( 1 – \frac{2i-2}{N-1} \right)^2 \right)^{-3/2} \) WMW、A131329 | ||||
| 0.607927101854026 | ハフナー=サルナック=マッカーリー定数 | Hafner–Sarnak–McCurley constant | 超 | |
| \(\displaystyle D \left( 1 \right) = \frac{6}{\pi^2} = \left( \zeta \left( 2 \right) \right)^{-1} = \prod_{p} \left( 1- p^{-2} \right) \) 互いに素である事に関したとある数式で現れる定数でもある。派生種複数あり WMW、A059956 | ||||
| 0.618033988749894 | 共役黄金比 | Golden ratio conjugate | 代 | |
| \(\displaystyle \varphi = \frac{\sqrt{5} -1}{2} \) WMW、A094214 | ||||
| 0.62432998854355 | ゴロム=ディックマン定数 | Golomb–Dickman constant | ||
| WMW、A084945 | ||||
| 0.62573580720527 | Kac Formula | |||
| \(\displaystyle C_{1} = \frac{2}{\pi} \left\{ \ln 2 + \int_{0}^{\infty} \left( \sqrt{x^{-2} – \frac{4e^{-2x}}{\left( 1-e^{-2x} \right)^2}} – \frac{1}{x+1} \right) dx \right\} \) WMW、A093601 | ||||
| 0.630929753571457 | カントール集合 | Fractal dimension of the Cantor set | 超 | |
| \(\displaystyle d_{f} \left( k \right) = \log_{3} 2 \) WMW、A102525 | ||||
| 0.632120558828557 | 時定数 | Time constant | 超 | |
| \(\displaystyle \tau = 1-e^{-1} \) Wiki | ||||
| 0.636619772367581 | ビュフォン定数 | Buffon constant | 超 | |
| \(\displaystyle \frac{2}{\pi} = \prod_{p \geq 3} \frac{p+2- \left( p \bmod 4 \right)}{p} \\\displaystyle = 1+ \sum_{n \geq 1} \left( -1 \right)^{n} \left( 4n+1 \right) \left( \frac{\prod_{i=1}^{n} 2i -1}{\prod_{j=1}^{n} 2j} \right) ^{3} \\\displaystyle = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2+ \sqrt{2}}}{2} \cdot \frac{\sqrt{2+ \sqrt{2+ \sqrt{2}}}}{2} \cdot \cdots \) 魔法幾何学的定数の値でもある OEIS、A060294 | ||||
| 0.643410546288338 | カエン定数 | Cahen’s constant | 超 | 4000 |
| \(\displaystyle \xi_{2} = C = \sum_{k \geq 1} \frac{\left( -1 \right)^{k}}{s_{k}-1} \\\displaystyle s_{0} = 2, s_{k+1} = 1+ \prod_{k=0}^{n} s_{k} = e_{k}^{2}-e_{k}+1 \) ⌊E2k+1⌋ WMW、A118227 | ||||
| 0.646245439894813 | Masser–Gramain constant | |||
| c=γβ(1)+β'(1) \(\displaystyle = \frac{\pi}{4} \left( 2 \gamma + 3 \ln \pi – 2 \ln 2 – 4 \ln \Gamma \left( 1/4 \right) \right) \) WMW、A086057 | ||||
| 0.658365599266331 | Lower límit iterated exponential | |||
| \(\displaystyle \lim_{n \to \infty} H_{2n} = \frac{1}{2} \uparrow \frac{1}{3} \uparrow \frac{1}{4} \uparrow \cdots \uparrow \frac{1}{2n} \) A242759 | ||||
| 0.660161815846869 | 双子素数定数 | twin primes constant | ||
| \(\displaystyle C_{2} = \prod_{p \geq 3} \frac{p \left( p-2 \right)}{\left( p-1 \right)^{2}} \) WMW、A005597 | ||||
| 0.661317049469622 | フェラー=トルニアー定数 | Feller–Tornier constant | 超? | |
| \(\displaystyle C_{FT} = \frac{1}{2} + \frac{1}{2} \prod_{p} \left( 1- \frac{2}{p^2} \right) \\\displaystyle = \frac{1}{2} + \frac{3}{\pi^2} \prod_{p} \left( 1- \frac{1}{p^2-1} \right) \\\displaystyle = \frac{1}{2} \left\{ 1+ \exp \left( -\sum_{n \geq 1} \frac{2^{n} P \left( n \right)}{n} \right) \right\} \) P(n)：素数ゼータ関数 WMW、A065493 | ||||
| 0.661707182267176 | Robbins constant | |||
| \(\displaystyle \Delta \left( 3 \right) = \frac{4+17 \sqrt{2} -6 \sqrt{3} -7 \pi}{105} + \frac{\ln \left( 1+ \sqrt{2} \right) + 2 \ln \left( 2+ \sqrt{3} \right)}{5} \) WMW、A073012 | ||||
| 0.662743419349181 | ラプラス極限定数 | Laplace limit constant | 超? | |
| \(\displaystyle \frac{x \exp \left( \sqrt{1+x^2} \right)}{1+\sqrt{1+x^2}} = 1 \) WMW、A033259 | ||||
| 0.670873 | Andrica予想 | Andrica’s Conjecture | ||
| √11-√7。連続する2つの素数p, qに対し、√q-√pが最大となる可能性のある値。 WMW | ||||
| 0.678234491917391 | 谷口定数 | Taniguchi’s constant | 超? | |
| \(\displaystyle C_{T} = \prod_{p} \left( 1- \frac{3}{p^3} + \frac{2}{p^4} + \frac{1}{p^5} – \frac{1}{p^6} \right) \\\displaystyle \exp \left( \sum_{n \geq 3} c_{n} P \left( n \right) \right) \\\displaystyle = \exp \left( – 3P \left( 3 \right) + 2P \left( 4 \right) + P \left( 5 \right) – \frac{11}{2} P \left( 6 \right) + 6P \left( 7 \right) + \cdots \right) \) WMW、A175639 | ||||
| 0.690347126114964 | Upper iterated exponential | |||
| \(\displaystyle \lim_{n \to \infty} H_{2n+1} = \frac{1}{2} \uparrow \frac{1}{3} \uparrow \frac{1}{4} \uparrow \cdots \uparrow \frac{1}{2n+1} \) A242760 | ||||
| 0.693147180559945 | ln(2) | 超 | ||
| 0.697774657964007 | 連分数定数 | Continued Fraction Constant | 無 | |
| \(\displaystyle C_1 = C_{CF} = \frac{I_{1} \left( 2 \right)}{I_{0} \left( 2 \right)} = \frac{\sum_{n \geq 0} \frac{n}{n!n!}}{\sum_{n \geq 0} \frac{1}{n!n!}} \\\displaystyle = K_{n \geq 1} \frac{1}{n} = \frac{1}{1+ \frac{1}{2+ \frac{1}{3+ \frac{1}{4+ \frac{1}{5+ \frac{1}{6+ \ddots}}}}}} \) WMW、A052119 | ||||
| 0.70258 | エンブリー=トレフェセン定数 | Embree–Trefethen constant | ||
| 乱数フィボナッチ数列の派生 WMW、A118288 | ||||
| 0.704442200999165 | Carefree constant | |||
| \(\displaystyle C_{2} = \prod_{p} \left(1- \frac{1}{p^2+p} \right) \) A065463 | ||||
| 0.7052301717918 | Primorial constant | 無 | ||
| \(\displaystyle \sum_{k \geq 1} \prod_{n=1}^{k} \frac{1}{p_{n}} \) A064648 | ||||
| 0.709803442861291 | Rabbit Constant | |||
| \(\displaystyle R = \sum_{n \geq 1} 2^{- \lfloor n \phi \rfloor} \) WMW、A014565 | ||||
| 0.714782700791294 | Traveling Salesman Constants | |||
| \(\displaystyle \lambda = \frac{ \left( 4 + 8 \sqrt{2} \right) \sqrt{51}}{153} \) 派生種にφ、ψがある WMW、A073008 | ||||
| 0.719960700043708 | Smarandache Constants | |||
| WMW | ||||
| 0.7236484022982 | サルナック定数 | Sarnak’s constant | 超? | |
| \(\displaystyle C_{s} = \prod_{p \geq 3} \left( 1- \frac{p+2}{p^3} \right) \) WMW、A065476 | ||||
| 0.737338303369284 | Grossman’s Constant | |||
| WMW、A085835 | ||||
| 0.739085133215 | ドッティ数 | Dottie number, Cosine, Inverse Cosine | 超 | |
| \(\displaystyle \cos \left( x \right) = x \\\displaystyle \lim_{n \to \infty} \cos^{n} \left( c \right) \\\displaystyle = \lim_{n \to \infty} \underbrace{ \cos \left( \cdots \cos \left( \cos \left( \cos \left( c \right) \right) \right) \cdots \right) }_{n}\) WMW、A003957 | ||||
| 0.740480489693061 | Hermite constant Sphere packing 3D Kepler conjecture | 超 | ||
| \(\displaystyle \mu_{K} = \frac{\pi}{3 \sqrt{2}} \) WMW、WMW、A093825 | ||||
| 0.747597920253411 | Rényi’s Parking Constant | |||
| \(\displaystyle m = \int_{0}^{\infty} \exp \left( -2 \int_{0}^{x} \frac{1-e^{-y}}{y} dy \right) dx \) WMW、A050996 | ||||
| 0.75576131407617 | icosahedron | |||
| Sqrt(42 + 18*sqrt(5))/12, 辺の長さが1の正二十面体に内接する球の半径？ A179294 | ||||
| 0.757823011268492 | Flajolet-Odlyzko Constant | |||
| WMW、A143297 | ||||
| 0.76422365358922 | ランダウ・ラマヌジャン定数 | Landau–Ramanujan constant | 超? | 30010 |
| \(\displaystyle K = \frac{1}{\sqrt{2}} \prod_{p \equiv 3 \bmod 4} \left( 1- p^{-2} \right)^{-1/2} \\\displaystyle = \frac{\pi}{4} \prod_{p \equiv 1 \bmod 4} \left( 1- p^{-2} \right)^{1/2} \) WMW、A064533 | ||||
| 0.765551370675726 | デラノイ数が関連する数 | Delannoy number | ||
| D(10^n)の桁数を小数展開した時の収束値 \(\displaystyle \log \left( 3 + 2 \sqrt{2} \right) \\\displaystyle \lim_{n \to \infty} 10^{-n} \sum_{k=0}^{10^{n}} { 10^{n} \choose k } { 10^{n} + k \choose k } \) WMW、A104178 | ||||
| 0.768225422326056 | Dedekind Eta Function | |||
| WMW、A091343 | ||||
| 0.779893400376822 | フレネル積分 | Fresnel integral | ||
| \(\displaystyle C \left( 1 \right) = \int_{0}^{1} \cos{\left( t^2 \right)} dt = \sum_{n \geq 0} \frac{\left( -1 \right)^{n}}{\left( 2n \right)! \left( 4n+1 \right)} \) 備考：\(\displaystyle C \left( x \right) = \int_{0}^{x} \cos{\left( t^2 \right)} dt = \sum_{n \geq 0} \left( -1 \right)^{n} \frac{x^{4n+1}}{\left( 2n \right)! \left( 4n+1 \right)} \) A099290 | ||||
| 0.783430510712134 | 二年生の夢 | Sophomore’s dream1 | ||
| \(\displaystyle I_{1} = \int_{0}^{1} x^{x} dx = \sum_{n \geq 1} \left( -1 \right)^{n+1} n^{-n} \) WMW、A083648 | ||||
| 0.788530565911508 | Lüroth constant | |||
| \(\displaystyle \sum_{n \geq 2} n^{-1} \ln \frac{n}{n-1} \) WMW、A085361 | ||||
| 0.794507192779479 | Kolakoski Constant | |||
| WMW、A118270 | ||||
| 0.809394020540639 | Alladi–Grinstead constant | |||
| \(\displaystyle A_{AG} = \exp \left( -1+ \sum_{k \geq 2} \sum_{n \geq 1} n^{-1} k^{-n-1} \right) \\\displaystyle = \exp \left( -1- \sum_{k \geq 2} k^{-1} \ln \left( 1-k^{-1} \right) \right) \) WMW、A085291 | ||||
| 0.812556559016006 | Stolarsky-Harborth Constant | |||
| WMW、A077464 | ||||
| 0.812557858821472 | Grothendieck’s Constant x0 | |||
| WMW、A088373 | ||||
| 0.822467033424113 | Nielsen–Ramanujan constant | 超 | ||
| \(\displaystyle \frac{\zeta \left( 2 \right)}{2} = \sum_{n \geq 1} \frac{\left( -1 \right)^{n+1}}{n^2} \) A072691 | ||||
| 0.8227 | Rutherford Constant | |||
| WMW | ||||
| 0.826993343132688 | Disk Covering | 超 | ||
| \(\displaystyle C_{5} = \left( \sum_{n \geq 0} {3n+2 \choose 2}^{-1} \right)^{-1} = \frac{3 \sqrt{3}}{2 \pi} \) WMW、A086089 | ||||
| 0.834626841674073 | ガウス定数 | Gauss’s constant | 超? | |
| \(\displaystyle G = \frac{1}{\mathrm{agm} \left( 1, \sqrt{2} \right) } = \frac{4 \sqrt{2} \left( \frac{1}{4}! \right)^2}{\pi^{3/2}} \\\displaystyle = \frac{2}{\pi} \int_{0}^{1} \frac{dx}{\sqrt{1-x^4}} \) agm(a, b)：算術幾何平均、派生種複数あり WMW、A014549 | ||||
| 0.835648848264721 | ベイカー定数 | Baker constant | ||
| \(\displaystyle \beta_{3} = \int_{0}^{1} \frac{dt}{1+t^3} = \sum_{n \geq 0} \frac{\left( -1 \right)^n}{3n+1} \\\displaystyle = \frac{1}{3} \left( \ln 2 + \frac{\pi}{\sqrt{3}} \right) \) A113476 | ||||
| 0.850736188201867 | Regular paperfolding sequence | |||
| \(\displaystyle P_{f} = \sum_{n \geq 0} \frac{8^{2^n}}{2^{2^{n+2}} -1} = \sum_{n \geq 0} \frac{2^{-2^n}}{1- 2^{-2^{n+2}}} \) WMW、A143347 | ||||
| 0.859099796854703 | Thue Constant | |||
| 0.11011011111011011111…2 Substitution Systemに基づく WMW、A074071 | ||||
| 0.862240125868054 | 2進数におけるチャンパーノウン定数 | Binary Champernowne Constant | ||
| C2=0.(1)(10)(11)(100)(101)(110)(111)…2 \(\displaystyle \sum_{n \geq 1} n \cdot 2^{-n – \sum_{k=1}^{n} \lfloor \log_{2} k \rfloor} \) WMW、A066716 | ||||
| 0.866025403784438 | ルベーグ最小値問題に関わる値の一つ | Lebesgue Minimal Problem | ||
| √3/2 WMW | ||||
| 0.87058838 | 四つ子素数におけるブルン定数 | Brun’s constant (Brun’s theorem) | 8 | |
| \(\displaystyle \sum_{p, p+2, p+6, p+8 \in \mathbb{ P }} \left( \frac{1}{p} + \frac{1}{p+2} + \frac{1}{p+6} + \frac{1}{p+8} \right) \) Wiki、A213007 | ||||
| 0.872284041065627 | Area of Ford circle | |||
| \(\displaystyle A_{CF} = \sum_{q \geq 1} \sum_{\left( p , q \right) = 1 \\ 1 \leq p \leq q} \frac{\pi}{4q^{4}} \\\displaystyle = \frac{\pi \zeta \left( 3 \right)}{4 \zeta \left( 4 \right)} = \frac{45 \zeta \left( 3 \right)}{2 \pi^3} \) WMW、A279037 | ||||
| 0.874464368404944 | 1と重複分を除く累乗数の逆数 | Perfect Power | ||
| \(\displaystyle \sum_{k \geq 2} \mu \left( k \right) \left[ 1 – \zeta \left( k \right) \right] \) μ(k)：メビウス関数 WMW、A072102 | ||||
| 0.88151383972517 | 二次類数定数 | Quadratic class number constant | ||
| \(\displaystyle Q = \prod_{p} \left( 1- \frac{1}{p^{3}+p^{2}} \right) \) WMW、A065465 | ||||
| 0.885603194410888 | ガンマ関数における関連する値 | Gamma Function | ||
| x>0 においてy = Γ(x)が最小になるyの値。この時のxの値は約1.461632144。 WMW、A030171 | ||||
| 0.908908557548541 | Complete Elliptic Integral | |||
| K(k) = 2E(k)を満たす値k WMW、A086199 | ||||
| 0.915502055389676 | Zolotarev-Schur Constant c | 超? | ||
| WMW、A143296 | ||||
| 0.915965594177219 | カタラン定数 | Catalan’s constant | 超? | 15510000000 |
| \(\displaystyle G = \beta \left( 2 \right) = \sum_{n \geq 0} \frac{\left( -1 \right)^{n}}{\left( 2n+1 \right)^{2}} \\\displaystyle = \int_{0}^{1} \int_{0}^{1} \frac{1}{1+x^{2} y^{2}} dxdy \) その他定積分の式が複数あり 備考(ディリクレ-ベータ関数)： \(\displaystyle \beta \left( s \right) = \sum_{n \geq 0} \frac{\left( -1 \right)^{n}}{\left( 2n+1 \right)^{s}} \) WMW、A006752 | ||||
| 0.923563831674181 | Mertens Theorem | |||
| \(\displaystyle \zeta \left( 2 \right) e^{- \gamma} \) WMW、A246499 | ||||
| 0.92883582713 | 双子素数の組の平均の逆数の合計 | Sum of the reciprocals of the averages of the twin prime pairs, JJGJJG | ||
| \(\displaystyle B_{1} = \frac{1}{4} + \frac{1}{6} + \frac{1}{12} + \frac{1}{18} + \frac{1}{30} + \frac{1}{42} + \cdots \) WMW、A241560、A014574 | ||||
| 0.955316618124509 | Magic angle | 超 | ||
| \(\displaystyle \theta_{m} = \arctan \left( \sqrt{2} \right) = \arccos \left( \sqrt{\frac{1}{3}} \right) \\\displaystyle \approx 54.7356^{\circ} \) Wiki、A195696 | ||||
| 0.970270114392033 | Lochs定数 | Lochs’s theorem | ||
| \(\displaystyle 6 \pi^{-2} \ln 2 \ln 10 \) WMW、A086819 | ||||
| 0.978012478186646 | Shapiro’s cyclic sum constant | |||
| WMW、A086278 | ||||
| 0.987700390736053 | ルーローの三角形 | Reuleaux triangle | 超 | |
| \(\displaystyle r^{2} \cdot \left( 2 \sqrt{3} + \frac{\pi}{6} -3 \right) : r^{2} \) 回転する半径rのルーローの三角形によって切り取られた領域の面積と一辺の長さrの正方形の面積比 WMW、A066666 | ||||
| 0.989431273831146 | Lebesgue constant c | |||
| \(\displaystyle \lim_{n \to \infty} \left( L_n – \frac{4}{\pi^2} \ln \left( 2n+1 \right) \right) \\\displaystyle = \frac{8}{\pi^2} \left( \sum_{k \geq 1} \frac{\ln k}{4k^2 -1} – \frac{\Gamma’ \left( 1/2 \right)}{\Gamma \left( 1/2 \right)} \right) \) リンク先に関連する定数複数あり WMW、A243277 | ||||
| 1 | 自 | ∞ | ||
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