数の比較Part04

2020.05.10
何のひねりもない数の比較の一覧表第4弾です。
今パートは1～√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自∞
\(\displaystyle 1 = \sum_{m \geq 2} \sum_{n \geq 2} m^{-n} \)
1.00743475688427DeVicci’s tesseract constant代
4x4-28x3-7x2+16x+16=0の解の一つ
A243309
1.01494160640965Gieseking constant
\(\displaystyle G = \int_{0}^{2 \pi/3} \ln \left( 2 \cos \frac{t}{2} \right) dt \\\displaystyle = 4^{-1} 3 \sqrt{3} \left( 1- \sum_{k \geq 0} \left( 3k-2 \right)^{-2} + \sum_{k \geq 1} \left( 3k+1 \right)^{-2} \right) \)
WMW、A143298
1.01734306198444ζ(6)Riemann zeta function超
\(\displaystyle \zeta \left( 6 \right) = \sum_{k \geq 1} k^{-6} = \frac{\pi^6}{945} \)
WMW、A013664
1.03465388189743Mertens Constant B2
\(\displaystyle B_{2} = \gamma + \sum_p \left[ \ln \left( 1 – p^{-1} \right) + \frac{1}{p – 1} \right] \\\displaystyle = B_{1} + \sum_p \frac{1}{p^{2} – p} \\\displaystyle = \gamma + \sum_{n \geq 2} \frac{\phi \left( n \right) \ln \left( \zeta \left( n \right) \right)}{n} \)
WMW、A083342
1.03692775514336ζ(5)Riemann zeta function
\(\displaystyle \zeta \left( 5 \right) = \sum_{k \geq 1} k^{-5} \)
WMW、A013663
1.05146222423826Golden Rhombus q
\(\displaystyle \frac{2}{\sqrt{1+ \phi^{2}}} = \csc \frac{2 \pi}{5} \\\displaystyle = \sqrt{2- \frac{2}{\sqrt{5}}} \)
WMW、A179290
1.0594630943592^(1/12)代
Wiki、A010774
1.082323ζ(4)Riemann zeta function超
\(\displaystyle \zeta \left( 4 \right) = \sum_{k \geq 1} k^{-4} = \frac{\pi^4}{90} \)
WMW、A013662
1.08366かつてのルジャンドル定数
現在は1に修正されている
WMW、A228211
1.09317045919549Smarandache Constant 1ª
派生種複数あり
WMW、A048799
1.09864196439415Paris Constant
\(\displaystyle C_{Pa} = \prod_{n \geq 2} \frac{2 \varphi}{\varphi + \varphi_{n}} \\\displaystyle \varphi = \frac{1+ \sqrt{5}}{2} , \varphi_{n} = \sqrt{1+ \varphi_{n-1}} , \varphi_{1} = 1 \)
WMW、A105415
1.09868580552518Lengyel’s Constant
WMW、A086053
1.1064957714de Bruijn Constant10?
WMW、A113276
1.10714871779409Golden Rhombus α
\(\displaystyle \cos^{-1} \frac{1}{\sqrt{5}} = \sec^{-1} \sqrt{5} = \sin^{-1} \frac{2}{\sqrt{5}} \\\displaystyle = \tan^{-1} 2 \approx 63.4349^{\circ} \)
WMW、A105199
1.11786415118994Goh-Schmutz constant
\(\displaystyle C_{GS} = \int_{0}^{\infty} \frac{ln \left( s+1 \right)}{e^{s}-1} ds \\\displaystyle = – \sum_{n \geq 1} n^{-1} e^{n} Ei \left( -n \right) \\\displaystyle = – \sum_{n \geq 1} n^{-1} e^{n} \left( \gamma + \ln n + \sum_{k \geq 1} \frac{n^k}{k \cdot k!} \right) \)
WMW、A143300
1.120529Golden Rectangle a
\(\displaystyle a = \left( 4/5 \right)^{1/4} \phi^{\left( \tan^{-1} 2 \right) / \pi} \)
WMW
1.1319882487943ヴィシュワナート定数Viswanath’s constant超?
乱数フィボナッチ数列の派生
WMW、A078416
1.13733873634419Tree Searching β
\(\displaystyle \beta = \sum_{k \geq 1} \left( 2^{k} – 1 \right)^{-2} \)
リンク先にこの定数の派生種あり
WMW、A065443
1.15470053837925Hermite constant代
\(\displaystyle \gamma_{2} = \frac{2}{\sqrt{3}} = \frac{1}{\cos \frac{\pi}{6}} \)
A020832
1.15636268433226Cubic recurrence constant
\(\displaystyle \sigma_{3} = \prod_{n \geq 1} n^{3^{-n}} \)
A123852
1.15872847301812山羊問題におけるレンズ形の面積Goat Problem
A=π/2、R=d=1のとき、
\begin{aligned}A={}&r^{2}\cos ^{-1}\left({\frac {d^{2}+r^{2}-R^{2}}{2dr}}\right)+R^{2}\cos ^{-1}\left({\frac {d^{2}-r^{2}+R^{2}}{2dR}}\right)\\&{}-{\frac {1}{2}}{\sqrt {(d+r-R)(d-r+R)(-d+r+R)(d+r+R)}} \end{aligned}
WMW、A133731
1.17082039324993Golden Rectangle x0
\(\displaystyle x_{0} = \sum_{n \geq 0} \left( \phi^{-4n} + \phi^{-4n-1} – \phi^{-4n-2} – \phi^{-4n-3} \right) \\\displaystyle = \frac{5 + 3 \sqrt{5}}{10} \)
WMW、A176015
1.17795605792266十二角数の逆数和dodecagonal number
\(\displaystyle \sum_{n \geq 1} \frac{1}{n \left( 5n-4 \right)} \\\displaystyle = 8^{-1} \pi \sqrt{1+ \frac{2}{\sqrt{5}}} + \frac{5}{16} \ln 5 + 8^{-1} \sqrt{5} \ln \left( \frac{1+ \sqrt{5}}{2} \right) \)
Wiki、A244649
1.17897974447216ギブズ現象による定数Wilbraham-Gibbs Constant
G = 2G’/π
WMW、A036793
1.18656911041562ヒンチン・レヴィ定数Khinchin–Lévy constant
\(\displaystyle \beta = \frac{\pi^2}{12 \ln 2} \)
WMW、A100199
1.1874523511265フォイアス定数αFoias Constant
\(\displaystyle x_{1} = \alpha , \quad x_{n+1} = \left( 1+\frac{1}{x_{n}} \right)^{n} \\\displaystyle \lim_{n \to \infty} x_{n} \frac{\ln n}{n} = \lim_{n \to \infty} \frac{x_{n}}{\pi \left( n \right)} = 1 \)
WMW、A085848
1.19704497以上のいとこ素数の逆数和Cousin Primes
\(\displaystyle \sum_{p, p+4 \in \mathbb{P} \\ p \geq 7} \left( \frac{1}{p} + \frac{1}{p+4} \right) \)
WMW
1.19967864025773下記の方程式の解LaplaceLimit, Hyperbolic Cotangent, Inverse Hyperbolic Cotangent
\(\displaystyle e^{x} \left( x-1 \right) = e^{-x} \left( x+1 \right) \)
WMW、WMW、A085984
1.20205690315959ζ(3)　(アペリーの定数)Apéry’s constant無15510000000
\(\displaystyle \zeta \left( 3 \right) = \sum_{k \geq 1} k^{-3} = \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{1}{1-xyz} dxdydz \\\displaystyle = \frac{1}{2} \sum_{i \geq 1} \sum_{j \geq 1} \frac{1}{ij \left( i+j \right) } \)
WMW、A002117
1.22541670246517Γ(3/4)
A068465
1.22674201072035Fibonacci Factorial constant
\(\displaystyle F = \prod_{n \geq 1} \left( 1- \left( -\varphi^{-2} \right)^{n} \right) \\\displaystyle = \prod_{n \geq 1} \left( 1- \left( \frac{\sqrt{5}-3}{2} \right)^{n} \right) \)
WMW、A062073
1.22834886703857qポッホハマー記号の最大値q-Pochhammer symbol
WMW、A143440
1.23370055013616K_2Favard constant超
\(\displaystyle \frac{3 \zeta \left( 2 \right)}{4} = \frac{\pi^2}{8} = \sum_{n \geq 0} \left( 2n-1 \right)^{-2} \)
A111003
1.25164759779046Bertrand’s constant b36?
2^(2^(2^b)) ≈ 37.0000000000944728917062132870071
2^(2^(2^(2^b))) ≈ 137438953481
A079614、A051501
1.259921049894872^(1/3)Delian Constant代
WMW、A002580
1.26185950714291Fractal dimension of the Koch snowflake超
\(\displaystyle C_{k} = \log_{3} 4 \)
WMW、A100831
1.2640847353053ヴァルディ定数Vardi’s constant
\(\displaystyle E = \frac{\sqrt{6}}{2} \exp \left\{ \sum_{j \geq 1} 2^{-j-1} \ln \left[ 1+ \left( 2e_{j} -1 \right)^{-2} \right] \right\} \\\displaystyle e_{0} = 2, e_{n} = 1+ \prod_{i=0}^{n-1} e_{i} = e_{n-1}^{2} – e_{n-1} +1 \)
WMW、A076393
1.27740905755963八角数の逆数和Octagonal Number
\(\displaystyle \sum_{n \geq 1} \frac{1}{n \left(3n-2 \right) } = \frac{\pi \sqrt{3} + 9 \ln{3}}{12} \)
WMW、A244645
1.28242712910062グレイシャー・キンケリン定数Glaisher–Kinkelin constant超?
\(\displaystyle A = e^{\frac{1}{12} – \zeta ‘ \left( -1 \right)} \\\displaystyle = \exp \left\{ \frac{1}{8} – \frac{1}{2} \sum_{n \geq 1} \frac{1}{n+1} \sum_{k=0}^{n} \left( -1 \right)^{k} {n \choose k} \left( k+1 \right)^{2} \ln \left( k+1 \right) \right\} \)
上記リンク先にこの定数の派生種あり
WMW、A074962
1.291285997二年生の夢Sophomore’s dream2
\(\displaystyle I_{2} = \int_{0}^{1} x^{-x} dx = \sum_{n \geq 1} n^{-n} \)
WMW、A073009
1.29751676555506Compositorial数の逆数和Compositorial numbers
\(\displaystyle a_n = \frac{c_k !}{c_k \sharp} = \prod_{i=1}^{n} c_i \)
でck：n番目の合成数、#：その数以下の素数階乗のとき、
\(\displaystyle \sum_{n \geq 1} a_{n}^{-1} \)
OEIS、Googology、A036691
1.30357726903429コンウェイ定数Conway’s constant代
\(\displaystyle x^{71} \quad -x^{69} -2x^{68} -x^{67} +2x^{66} +2x^{65} +x^{64} \\\displaystyle -x^{63} -x^{62} -x^{61} -x^{60} -x^{59} +2x^{58} +5x^{57} +3x^{56} \\\displaystyle -2x^{55} -10x^{54} -3x^{53} -2x^{52} +6x^{51} +6x^{50} +x^{49} +9x^{48} \\\displaystyle -3x^{47} -7x^{46} -8x^{45} -8x^{44} +10x^{43} +6x^{42} +8x^{41} -5x^{40} \\\displaystyle -12x^{39} +7x^{38} -7x^{37} +7x^{36} +x^{35} -3x^{34} +10x^{33} +x^{32} \\\displaystyle -6x^{31} -2x^{30} -10x^{29} -3x^{28} +2x^{27} +9x^{26} -3x^{25} +14x^{24} \\\displaystyle -8x^{23} \quad -7x^{21} +9x^{20} +3x^{19} -4x^{18} -10x^{17} -7x^{16} \\\displaystyle +12x^{15} +7x^{14} +2x^{13} -12x^{12} -4x^{11} -2x^{10} +5x^{9} \quad \\\displaystyle +x^{7} -7x^{6} +7x^{5} -4x^{4} +12x^{3} -6x^{2} +3x^{1} -6 = 0 \)
WMW、A014715
1.30568672804987Fractal dimension of the Apollonian packing of circles
A052483
1.30637788386308ミルス定数AMills’ constant6850
\(\displaystyle \lfloor A ^{3^n} \rfloor \in \mathbb{P} \)
リーマン仮説が真の場合を想定
WMW、A051021
1.306951メルゲルヤンの定理に関わる定数Mergelyan’s Theorem
WMW
1.311028777146051st lemniscate constant
L1 = L/2
WMW、A085565
1.32032363169373Cousin Primes
関連
WMW、A114907
1.32277925312238七角数の逆数和Heptagonal numbers
\(\displaystyle \sum_{n \geq 1} \frac{2}{n \left( 5n-3 \right) } \\\displaystyle = \frac{\pi}{15} { \sqrt{25-10 \sqrt{5}}} + \frac{2}{3} \ln \left( 5 \right) + \frac{{1} + \sqrt{5}}{3} \ln \left( \frac{1}{2} \sqrt{10-2 \sqrt{5}} \right) \\\displaystyle + \frac{{1}- \sqrt{5}}{3} \ln \left( \frac{1}{2} \sqrt{10+2 \sqrt{5}} \right) \)
WMW、A244639
1.32471795724474プラスチック数 (x^3 = x+1)Plastic number代
\(\displaystyle \sqrt[3]{1+ \sqrt[3]{1+ \sqrt[3]{1+ \cdots}}} \\\displaystyle \sqrt[3]{\frac{1}{2} + \frac{\sqrt{69}}{18}} + \sqrt[3]{\frac{1}{2} – \frac{\sqrt{69}}{18}} \)
Pisot数にも関連あり
WMW、A060006
1.33258227573322Mertens Constant B3
\(\displaystyle B_{3} = \gamma + \sum_{j \geq 2} \sum_p \frac{\ln p}{p^j} \\\displaystyle = \lim_{x \to \infty} \left( \ln x – \sum_{p \leq x} \frac{\ln p}{p} \right) \)
WMW、A083343
1.33978415357434トーシェント総和定数Totient summatory constant
\(\displaystyle \prod_{p} \left( 1+ \frac{1}{p^3 – p^2} \right) \)
リンク先及びA000010に関連する定数が複数あり
WMW、A065483
1.35113157449165Vallée Constant
派生種。
\(\displaystyle \langle N \rangle = \frac{3}{4} + \frac{180}{\pi^{4}} \sum_{i \geq 1} \sum_{j=i+1}^{2i} \frac{1}{i^{2} j^{2}} \\\displaystyle = \frac{17}{4} + \frac{360}{\pi^{4}} \sum_{i \geq 1} \sum_{j=1}^{i} \frac{\left( -1 \right)^{i}}{i^{2} j^{2}} \\\displaystyle = 17- \frac{60}{\pi^{4}} \left( 24 \sum_{k \geq 1} 2^{-k} k^{4} – \pi^{2} \ln^{2} 2 + 21 \zeta \left( 3 \right) \ln 2 + \ln^{4} 2 \right) \)
WMW、A074903
1.35845627418298Golden spiral
\(\displaystyle c = \varphi^{2/\pi} = \left( \frac{1+ \sqrt{5}}{2} \right)^{2/\pi} \)
Wiki、A212224
1.38027756909761Pisot-Vijayaraghavan Constant
x4 – x3 – 1 = 0の解の一つ
WMW、A086106
1.38629436111989六角数の逆数和Hexagonal number
\(\displaystyle \sum_{n \geq 1} \frac{1}{n \left( 2n-1 \right)} = \)
(中略)
\(\displaystyle = \ln 4 \)
WMW、A016627
1.3954859724793Hard Hexagon Entropy Constant代
方程式と解はリンク先参照
WMW、A085851
1.40457593466374Grothendieck’s Constant
\(\displaystyle \left( \int_{0}^{\pi/2} \frac{\cos^{2} \theta}{\sqrt{1 + \sin^{2} \theta}} d \theta \right)^{-1} \\\displaystyle = \frac{1}{2K \left( i \right) – E \left( i \right)} \)
→K(i)：第1種完全楕円積分、E(i)：第2種完全楕円積分
WMW、A088375
1.40490913273579Grothendieck’s Constant kC
WMW、A088374
1.41421356237309√2代10兆
WMW

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