Mathematical constants and functions

A mathematical constant is a number, which has a special meaning for calculations. For example, the constant π means the ratio of the length of a circle's circumference to its diameter. This value is always the same for any circle.

Tables structure

• Value numerical of the constant and link to MathWorld or to OEIS Wiki.
• LaTeX: Formula or series in TeX format.
• Formula: For use in program Wolfram Alpha.
• OEIS: On-Line Encyclopedia of Integer Sequences.
• Continued fraction: In the simple form [to integer; frac1, frac2, frac3, ...], overline if periodic.
• Year: Discovery of the constant, or dates of the author.
• Web format: Value in appropriate format for web browsers.
• Nº: Number types.
  • R – Rational number
  • I – Irrational number
  • A – Algebraic number
  • T – Transcendental number
  • C – Non-real Complex number

Table of constants and functions

You can choose the order of the list by clicking on the name, value, OEIS, etc..

Value	Name	Graphics	Symbol	LaTeX	Formula	Nº	OEIS	Continued fraction	Year	Web format
0.74048 04896 93061 04116	Hermite constant Sphere packing 3D Kepler conjecture	100px	${\mu_{_{K}}}$	$\frac{\pi}{3\sqrt{2}} {\color{white}......\color{black}}$ The Flyspeck project, led by Thomas Hales, demonstrated in 2014 that Kepler's conjecture is true.	pi/(3 sqrt(2))		A093825	[0;1,2,1,5,1,4,2,2,1,1,2,2,2,6,1,1,1,5,2,1,1,1, ...]	1611	0.74048048969306104116931349834344894
22.45915 77183 61045 47342	pi^e		$\pi^{e}$	$\pi^{e}$	pi^e		A059850	[22;2,5,1,1,1,1,1,3,2,1,1,3,9,15,25,1,1,5,...]		22.4591577183610454734271522045437350
2.80777 02420 28519 36522	Fransén-Robinson constant		${F}$	$\int_{0}^\infty \frac{1}{\Gamma(x)}\, dx. = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, dx$	N[int[0 to ∞] {1/Gamma(x)}]		A058655	[2;1,4,4,1,18,5,1,3,4,1,5,3,6,1,1,1,5,1,1,1...]	1978	2.80777024202851936522150118655777293
1.30568 6729 ≈ by Thomas & Dhar 1.30568 8 ≈ by McMullen	Fractal dimension of the Apollonian packing of circles  ·	100px	$\varepsilon$				A052483	[0;3,2,3,16,8,10,3,1,1,2,1,3,1,2,13,1,1,4,1,5,...]	1994 1998	1.305686729 ≈ 1.305688 ≈
0.43828 29367 27032 11162 0.36059 24718 71385 485 i	Infinite Tetration of i		${}^\infty {i}$	$\lim_{n \to \infty} {}^n i = \lim_{n \to \infty} \underbrace{i^{i^{\cdot^{\cdot^{i}}}}}_n$	i^i^i^i^i^i^...	C	A077589  A077590	[0;2,3,1,1,4,2,2,1,10,2,1,3,1,8,2,1,2,1, ...] + [0;2,1,3,2,2,3,1,5,5,1,2,1,10,10,6,1,1...] i		0.43828293672703211162697516355126482 + 0.36059247187138548595294052690600 i
0.92883 58271	Sum of the reciprocals of the averages of the twin prime pairs, JJGJJG		$B_1$	$\frac{1}{4}+\frac{1}{6}+\frac{1}{12}+\frac{1}{18}+\frac{1}{30}+\frac{1}{42}+\frac{1}{60}+\frac{1}{72}+\cdots$	1/4 + 1/6 + 1/12 + 1/18 + 1/30 + 1/42 + 1/60 + 1/72 + ...		A241560	[0; 1, 13, 19, 4, 2, 3, 1, 1]	2014	0.928835827131
0.63092 97535 71457 43709	Fractal dimension of the Cantor set	100px	$d_f(k)$	$\lim_{\varepsilon \to 0} \frac {\log N(\varepsilon)}{\log (1/\varepsilon)} = \frac{\log 2}{\log 3}$	log(2)/log(3) N[3^x=2]	T	A102525	[0;1,1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...]		0.63092975357145743709952711434276085
0.31830 98861 83790 67153	Inverse of Pi, Ramanujan		$\frac{1}{\pi}$	$\frac{2\sqrt{2}}{9801} \sum^\infty_{n=0} \frac{(4n)!\,(1103+26390 \; n)}{(n!)^4 \, 396^{4n}}$	2 sqrt(2)/9801 * Sum[n=0 to ∞] {((4n)!/n!^4) *(1103+ 26390n) / 396^(4n)}	T	A049541	[0;3,7,15,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...]		0.31830988618379067153776752674502872
0.28878 80950 86602 42127	Flajolet and Richmond		${Q}$	$\prod_{n=1}^{\infty} \left(1 - \frac{1}{2^n}\right) = \left(1{-}\frac{1}{2^1}\right) \left(1{-}\frac{1}{2^2} \right)\left(1{-}\frac{1}{2^3} \right) ...$	prod[n=1 to ∞] {1-1/2^n}		A048651	[0;3,2,6,4,1,2,1,9,2,1,2,3,2,3,5,1,2,1,1,6,1,...]	1992	0.28878809508660242127889972192923078
1.53960 07178 39002 03869	Lieb's square ice constant	100px	${W}_{2D}$	$\lim_{n\to\infty}(f(n))^{n^{-2}}=\left(\frac{4}{3}\right)^\frac{3}{2}=\frac{8}{3\sqrt3}$	(4/3)^(3/2)	A	A118273	[1;1,1,5,1,4,2,1,6,1,6,1,2,4,1,5,1,1,2,...]	1967	1.53960071783900203869106341467188655
0.20787 95763 50761 90854	$i^i$		$i^i$	$e^{-\frac{\pi}{2}}$	e^(-π/2)	T	A049006	[0;4,1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,...]	1746	0.20787957635076190854695561983497877
4.53236 01418 27193 80962	Van der Pauw constant		${\alpha}$	$\frac{\pi}{\ln(2)}=\frac{\sum\limits_{n=0}^\infty \frac{4(-1)^n}{2n+1}}{\sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n}}=\frac{\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\cdots}{\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots}$	π/ln(2)		A163973	[4;1,1,7,4,2,3,3,1,4,1,1,4,7,2,3,3,12,2,1,...]		4.53236014182719380962768294571666681
0.76159 41559 55764 88811	Hyperbolic tangent of 1		${th} \, 1$	$-i \tan (i) = \frac{e-\frac{1}{e}}{e+\frac{1}{e}} = \frac{e^2-1}{e^2+1}$	(e-1/e)/(e+1/e)	T	A073744	[0;1,3,5,7,9,11,13,15,17,19,21,23,25,27,...] = [0;2p+1], p∈ℕ		0.76159415595576488811945828260479359
0.59017 02995 08048 11302	Chebyshev constant  ·		${\lambda_{Ch}}$	$\frac{\Gamma(\tfrac14)^2}{4 \pi^{3/2}} = \frac{4 (\tfrac14 !)^2}{\pi^{3/2}}$	(Gamma(1/4)^2) /(4 pi^(3/2))		A249205	[0;1,1,2,3,1,2,41,1,6,5,124,5,2,2,1,1,6,1,2,...]		0.59017029950804811302266897027924429
0.07077 60393 11528 80353 -0.68400 03894 37932 129 i	MKB constant  ·  ·		$M_I$	$\lim_{n\rightarrow \infty} \int_{1}^{2n} (-1)^x ~ \sqrt[x]{x} ~ dx = \int_{1}^{2n} e^{i \pi x} ~ x^{1/x} ~ dx$	lim_(2n->∞) int[1 to 2n] {exp(i*Pi*x)*x^(1/x) dx}	C	A255727  A255728	[0;14,7,1,2,1,23,2,1,8,16,1,1,3,1,26,1,6,1,1, ...] - [0;1,2,6,13,41,112,1,25,1,1,1,1,3,13,2,1, ...] i	2009	0.07077603931152880353952802183028200 -0.68400038943793212918274445999266 i
1.25992 10498 94873 16476	Cube root of 2 Delian Constant	100px	$\sqrt[3]{2}$	$\sqrt[3]{2}$	2^(1/3)	A	A002580	[1;3,1,5,1,1,4,1,1,8,1,14,1,10,2,1,4,12,2,3,...]		1.25992104989487316476721060727822835
1.09317 04591 95490 89396	Smarandache Constant 1ª		${S_1}$	$\sum_{n=2}^\infty \frac1{\mu(n)!} {\color{white}....\color{black}}$ where μ(n) is the Kempner function			A048799	[1;10,1,2,1,2,1,13,3,1,6,1,2,11,4,6,2,15,1,1,...]		1.09317045919549089396820137014520832
0.62481 05338 43826 58687 + 1,30024 25902 20120 419 i	Generalized continued fraction of i		${{F}_{CG}}_{(i)}$	$\textstyle i{+}\frac i{i+\frac i{i+\frac i{i+\frac i{i+\frac i{i+\frac i{i+i{/...}}}}}}} = \sqrt{\frac{\sqrt{17}-1}{8}} + i \left(\tfrac12 {+} \sqrt{\frac{2}{\sqrt{17}-1}}\right)$	i+i/(i+i/(i+i/(i+i/(i+i/( i+i/(i+i/(i+i/(i+i/(i+i/( i+i/(i+i/(i+i/(i+i/(i+i/( i+i/(i+i/(i+i/(i+i/(i+i/( ...)))))))))))))))))))))	C A	A156590  A156548	[i;1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,..] = [0;1,i]		0.62481053384382658687960444744285144 + 1.30024259022012041915890982074952 i
3.05940 74053 42576 14453	Double factorial constant	100px	${C_{_{n!!}}}$	$\sum_{n=0}^{\infty} \frac{1}{n!!} = \sqrt{e} \left[\frac {1}{\sqrt{2}}+\gamma(\tfrac12 ,\tfrac12)\right]$	Sum[n=0 to ∞]{1/n!!}		A143280	[3;16,1,4,1,66,10,1,1,1,1,2,5,1,2,1,1,1,1,1,2,...]		3.05940740534257614453947549923327861
5.97798 68121 78349 12266	Madelung Constant 2		${H}_{2}(2)$	$\pi \ln(3) \sqrt 3$	Pi Log[3]Sqrt[3]		A086055	[5;1,44,2,2,1,15,1,1,12,1,65,11,1,3,1,1,...]		5.97798681217834912266905331933922774
0.91893 85332 04672 74178	Raabe's formula		${\zeta'(0)}$	$\int\limits_a^{a+1}\log\Gamma(t)\,\mathrm dt = \tfrac12\log2\pi + a\log a - a,\quad a \ge 0$	integral_a^(a+1) {log(Gamma(x))+a-a log(a)} dx		A075700	[0;1,11,2,1,36,1,1,3,3,5,3,1,18,2,1,1,2,2,1,1,...]		0.91893853320467274178032973640561763
2.20741 60991 62477 96230	Lower limit in the moving sofa problem	100px	${S_{_{H}}}$	$\frac {\pi}{2} +\frac {2}{\pi} \,$	pi/2 + 2/pi	T	A086118	[2;4,1,4,1,1,2,5,1,11,1,1,5,1,6,1,3,1,1,1,1,7,...]	1967	2.20741609916247796230685674512980889
1.17628 08182 59917 50654	Salem number, Lehmer's conjecture		${\sigma_{_{10}}}$	$x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$	x^10+x^9-x^7-x^6 -x^5-x^4-x^3+x+1	A	A073011	[1;5,1,2,17,1,7,2,1,1,2,4,7,2,2,1,1,15,1,1, ...	1983?	1.17628081825991750654407033847403505
0.37395 58136 19202 28805	Artin constant		${C}_{Artin}$	$\prod_{n=1}^{\infty} \left(1-\frac{1}{p_n(p_n-1)}\right)\quad p_n \scriptstyle \text{ = prime}$	Prod[n=1 to ∞] {1-1/(prime(n) (prime(n)-1))}		A005596	[0;2,1,2,14,1,1,2,3,5,1,3,1,5,1,1,2,3,5,46,...]	1999	0.37395581361920228805472805434641641
0.42215 77331 15826 62702	Volume of Reuleaux tetrahedron	100px	${V_{_{R}}}$	$\frac{s^3}{12}(3\sqrt2 - 49 \, \pi + 162 \, \arctan\sqrt2)$	(3*Sqrt[2] - 49*Pi + 162*ArcTan[Sqrt[2]])/12		A102888	[0;2,2,1,2,2,7,4,4,287,1,6,1,2,1,8,5,1,1,1,1, ...]		0.42215773311582662702336591662385075
2.82641 99970 67591 57554	Murata Constant		${C_m}$	$\prod_{n = 1}^\infty \underset{p_{n}: \, {prime}}{ \Big(1 + \frac{1}{(p_n-1)^2}\Big)}$	Prod[n=1 to ∞] {1+1/(prime(n) -1)^2}		A065485	[2;1,4,1,3,5,2,2,2,4,3,2,1,3,2,1,1,1,8,2,2,28,...]		2.82641999706759157554639174723695374
1.09864 19643 94156 48573	Paris Constant		$C_{Pa}$	$\prod_{n=2}^\infty \frac{2 \varphi}{\varphi+ \varphi_n} \; ,\; \varphi {=} {Fi}$  con  $\varphi_n {=} \sqrt{1 {+} \varphi_{n {-} 1}}$   y   $\varphi_1 {=} 1$			A105415	[1;10,7,3,1,3,1,5,1,4,2,7,1,2,3,22,1,2,5,2,1,...]		1.09864196439415648573466891734359621
2.39996 32297 28653 32223 Radians	Golden angle	50px	${b}$	$(4-2\,\Phi)\,\pi = (3-\sqrt{5})\,\pi$ = 137.5077640500378546 ...°	(4-2*Phi)*Pi	T	A131988	[2;2,1,1,1087,4,4,120,2,1,1,2,1,1,7,7,2,11,...]	1907	2.39996322972865332223155550663361385
1.64218 84352 22121 13687	Lebesgue constant L2		${L2}$	$\frac{1}{5} + \frac{\sqrt{25-2\sqrt{5}}}{\pi} = \frac{1}{\pi} \int_0^\pi \frac {\left|\sin(\frac{5t}{2})\right|} {\sin(\frac{t}{2})} \,d t$	1/5 + sqrt(25 - 2*sqrt(5))/Pi	T	A226655	[1;1,1,1,3,1,6,1,5,2,2,3,1,2,7,1,3,5,2,2,1,1,...]	1910	1.64218843522212113687362798892294034
1.26408 47353 05301 11307	Vardi constant		${V_c}$	$\frac{\sqrt{3}}{\sqrt{2}}\prod_{n\ge1}\left(1+{1\over(2e_n-1)^2}\right)^{\!1/2^{n+1}}$			A076393	[1;3,1,3,1,2,5,54,7,1,2,1,2,3,15,1,2,1,1,2,1,...]	1991	1.26408473530530111307959958416466949
1.5065918849 ± 0.0000000028	Area of the Mandelbrot fractal	100px	$\gamma$	This is conjectured to be: $\sqrt{6\pi -1} - e = 1.506591651\cdots$			A098403	[1;1,1,37,2,2,1,10,1,1,2,2,4,1,1,1,1,5,4,...]	1912	1.50659177 +/- 0.00000008
1.61111 49258 08376 736 111···111 27224 36828 183213 ones	Exponential factorial constant		${S_{Ef}}$	$\sum_{n=1}^{\infty} \frac{1}{n^{(n{-}1)^{\cdot^{\cdot^{\cdot^{2^1}}}}}} = 1 {+} \frac{1}{2^{1}} {+} \frac{1}{3^{2^{1}}} + \frac{1}{4^{3^{2^{1}}}} + \frac{1}{5^{4^{3^{2^{1}}}}} {+} \cdots$		T	A080219	[1; 1, 1, 1, 1, 2, 1, 808, 2, 1, 2, 1, 14,...]		1.61111492580837673611111111111111111
1.11786 41511 89944 97314	Goh-Schmutz constant		$C_{GS}$	$\int^\infty_0\frac{\log(s+1)}{e^s-1} \ ds = \! - \! \sum_{n=1}^\infty \frac {e^n}{n} Ei(-n)$ Ei: Exponential Integral	Integrate{ log(s+1) /(E^s-1)}		A143300	[1;8,2,15,2,7,2,1,1,1,1,2,3,5,3,5,1,1,4,13,1,...]		1.11786415118994497314040996202656544
0.31813 15052 04764 13531 ±1.33723 57014 30689 40 i	Fixed points Super-Logarithm · Tetration	100px	${-W(-1)}$	$\lim_{n\rightarrow \infty}$$f(x) = \underbrace{\log(\log(\log(\log(\cdots\log(\log(x)))))) \,\! }\atop {\log_s \text{ n times}}$ For an initial value of x different to ${\textstyle 0, 1, e, e^e, e^{e^e} }$, etc.	-W(-1) where W=ProductLog Lambert W function	C	A059526  A059527	[-i;1 +2i,1+i,6-i,1+2i,-7+3i,2i,2,1-2i,-1+i,-, ...]		0.31813150520476413531265425158766451 -1.33723570143068940890116214319371 i
0.28016 94990 23869 13303	Bernstein's constant		${\beta}$	$\approx \frac {1}{2\sqrt {\pi}}$	1/(2 sqrt(pi))	T	A073001	[0;3,1,1,3,9,6,3,1,3,14,34,2,1,1,60,2,2,1,1,...]	1913	0.28016949902386913303643649123067200
0.66016 18158 46869 57392	Twin Primes Constant		${C}_{2}$	$\prod_{p=3}^\infty \frac{p(p-2)}{(p-1)^2}$	prod[p=3 to ∞] {p(p-2)/(p-1)^2		A005597	[0;1,1,1,16,2,2,2,2,1,18,2,2,11,1,1,2,4,1,...]	1922	0.66016181584686957392781211001455577
1.22674 20107 20353 24441	Fibonacci Factorial constant		$F$	$\prod_{n = 1}^\infty \left(1 - \left( -\frac{1}{{\varphi}^2}\right)^n \right)= \prod_{n = 1}^\infty \left(1 - \left( \frac{\sqrt{5}-3}{2}\right)^n \right)$	prod[n=1 to ∞] {1-((sqrt(5) -3)/2)^n}		A062073	[1;4,2,2,3,2,15,9,1,2,1,2,15,7,6,21,3,5,1,23,...]		1.22674201072035324441763023045536165
0.11494 20448 53296 20070	Kepler–Bouwkamp constant	100px	${\rho}$	$\prod_{n=3}^\infty \cos\left(\frac{\pi}{n} \right) = \cos\left(\frac{\pi}{3} \right) \cos\left(\frac{\pi}{4} \right) \cos\left(\frac{\pi}{5}\right) ...$	prod[n=3 to ∞] {cos(pi/n)}		A085365	[0;8,1,2,2,1,272,2,1,41,6,1,3,1,1,26,4,1,1,...]		0.11494204485329620070104015746959874
1.78723 16501 82965 93301	Komornik–Loreti constant		${q}$	$1 = \!\sum_{n=1}^\infty \frac{t_k}{q^k} \qquad \scriptstyle \text{Raiz real de} \displaystyle\prod_{n=0}^\infty \!\left (\! 1 {-} \frac{1}{q^{2^n}} \!\right ) \! {+} \frac{q{-}2}{q{-}1}=0$ tk = Thue–Morse sequence	FindRoot[(prod[n=0 to ∞] {1-1/(x^2^n)}+(x-2) /(x-1))= 0, {x, 1.7}, WorkingPrecision->30]	T	A055060	[1;1,3,1,2,3,188,1,12,1,1,22,33,1,10,1,1,7,...]	1998	1.78723165018296593301327489033700839
3.30277 56377 31994 64655	Bronze ratio		${\sigma}_{\,Rr}$	$\frac {3+\sqrt{13}}{2} = 1+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}}$	(3+sqrt 13)/2	A	A098316	[3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...] = [3;3,...]		3.30277563773199464655961063373524797
0.82699 33431 32688 07426	Disk Covering	100px	${C_5}$	${\frac{1}{{\sum\limits_{n=0}^\infty \frac{1}{\binom{3n+2}{2}}}}}=\frac{3\sqrt3}{2\pi}$	3 Sqrt[3]/(2 Pi)	T	A086089	[0;1,4,1,3,1,1,4,1,2,2,1,1,7,1,4,4,2,1,1,1,1,...]	1939 1949	0.82699334313268807426698974746945416
2.66514 41426 90225 18865	Gelfond–Schneider constant		$G_{\,GS}$	$2^{\sqrt{2}}$	2^sqrt{2}	T	A007507	[2;1,1,1,72,3,4,1,3,2,1,1,1,14,1,2,1,1,3,1,...]	1934	2.66514414269022518865029724987313985
3.27582 29187 21811 15978	Khinchin-Lévy constant		$\gamma$	$e^{\pi^2/(12\ln2)}$	e^(\pi^2/(12 ln(2))		A086702	[3;3,1,1,1,2,29,1,130,1,12,3,8,2,4,1,3,55,...]	1936	3.27582291872181115978768188245384386
0.52382 25713 89864 40645	Chi Function Hyperbolic cosine integral	100px	${\operatorname{Chi()}}$	$\gamma + \int_0^x\frac{\cosh t-1}{t}\,dt$ $\scriptstyle \gamma \, \text{= Euler–Mascheroni constant= 0.5772156649...}$	Chi(x)		A133746	[0;1,1,9,1,172,1,7,1,11,1,1,2,1,8,1,1,1,1,1,...]		0.52382257138986440645095829438325566
1.13198 82487 943	Viswanath constant		${C}_{Vi}$	$\lim_{n \to \infty}|a_n|^\frac{1}{n}$      where an = Fibonacci sequence	lim_(n->∞) |a_n|^(1/n)	T ?	A078416	[1;7,1,1,2,1,3,2,1,2,1,8,1,5,1,1,1,9,1,...]	1997	1.1319882487943 ...
1.23370 05501 36169 82735	Favard constant		$\tfrac34\zeta(2)$	$\frac{\pi^2}{8} = \sum_{n = 0}^\infty \frac{1}{(2n-1)^2} = \frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots$	sum[n=1 to ∞] {1/((2n-1)^2)}	T	A111003	[1;4,3,1,1,2,2,5,1,1,1,1,2,1,2,1,10,4,3,1,1,...]	1902 a 1965	1.23370055013616982735431137498451889
2.50662 82746 31000 50241	Square root of 2 pi	100px	$\sqrt{2 \pi}$	$\sqrt{2 \pi} = \lim_{n \to \infty} \frac {n! \; e^n}{n^n \sqrt{n}}{\color{white}....\color{black}}$ Stirling's approximation	sqrt (2 pi)	T	A019727	[2;1,1,37,4,1,1,1,1,9,1,1,2,8,6,1,2,2,1,3,...]	1692 a 1770	2.50662827463100050241576528481104525
4.13273 13541 22492 93846	Square root of Tau·e		$\sqrt{\tau e}$	$\sqrt{2 \pi e}$	sqrt(2 pi e)		A019633	[4;7,1,1,6,1,5,1,1,1,8,3,1,2,2,15,2,1,1,2,4,...]		4.13273135412249293846939188429985264
0.97027 01143 92033 92574	Lochs constant		${\text{£}_{_{Lo}}}$	$\frac {6 \ln 2 \ln 10}{ \pi^2}$	6*ln(2)*ln(10)/Pi^2		A086819	[0;1,32,1,1,1,2,1,46,7,2,7,10,8,1,71,1,37,1,1,...]	1964	0.97027011439203392574025601921001083
0.98770 03907 36053 46013	Area bounded by the eccentric rotation of Reuleaux triangle		$\mathcal{T}_R$	$a^2 \cdot \left( 2\sqrt{3} + {\frac{\pi}{6}} - 3 \right)$    where a= side length of the square	2 sqrt(3)+pi/6-3	T	A066666	[0;1,80,3,3,2,1,1,1,4,2,2,1,1,1,8,1,2,10,1,2,...]	1914	0.98770039073605346013199991355832854
0.70444 22009 99165 59273	Carefree constant 2		$\mathcal{C}_2$	$\underset{ p_n: \, {prime}}{\prod_{n = 1}^\infty \left(1 - \frac{1}{p_n(p_n+1)}\right)}$	N[prod[n=1 to ∞] {1 - 1/(prime(n)* (prime(n)+1))}]		A065463	[0;1,2,2,1,1,1,1,4,2,1,1,3,703,2,1,1,1,3,5,1,...]		0.70444220099916559273660335032663721
1.84775 90650 22573 51225	Connective constant	100px	${\mu}$	$\sqrt{2 + \sqrt{2}} \; = \lim_{n \rightarrow \infty} c_n^{1/n}$ as a root of the polynomial $: \; x ^ 4-4 x ^ 2 + 2=0$	sqrt(2+sqrt(2))	A	A179260	[1;1,5,1,1,3,6,1,3,3,10,10,1,1,1,5,2,3,1,1,3,...]		1.84775906502257351225636637879357657
0.30366 30028 98732 65859	Gauss-Kuzmin-Wirsing constant		${\lambda}_{2}$	$\lim_{n \to \infty}\frac{F_n(x) - \ln(1 - x)}{(-\lambda)^n} = \Psi(x),$ where $\Psi(x)$ is an analytic function with $\Psi(0) \!=\! \Psi(1) \!=\! 0$.			A038517	[0;3,3,2,2,3,13,1,174,1,1,1,2,2,2,1,1,1,2,2,1,...]	1973	0.30366300289873265859744812190155623
1.57079 63267 94896 61923	Favard constant K1 Wallis product	100px	${\frac{\pi}{2}}$	$\prod_{n=1}^{\infty} \left(\frac{4n^2}{4n^2 - 1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots$	Prod[n=1 to ∞] {(4n^2)/(4n^2-1)}	T	A069196	[1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,1,5,1...]	1655	1.57079632679489661923132169163975144
1.60669 51524 15291 76378	Erdős–Borwein constant		${E}_{\,B}$	$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}\frac{1}{2^{mn}} =\sum_{n=1}^{\infty}\frac{1}{2^n-1} = \frac{1}{1} \! + \! \frac{1}{3} \! + \! \frac{1}{7} \! + \! \frac{1}{15} \! + \! ...$	sum[n=1 to ∞] {1/(2^n-1)}	I	A065442	[1;1,1,1,1,5,2,1,2,29,4,1,2,2,2,2,6,1,7,1,...]	1949	1.60669515241529176378330152319092458
1.61803 39887 49894 84820	Phi, Golden ratio	100px	${\varphi}$	$\frac{1 + \sqrt{5}}{2} = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}$	(1+5^(1/2))/2	A	A001622	[0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...] = [0;1,...]	-300 ~	1.61803398874989484820458683436563811
1.64493 40668 48226 43647	Riemann Function Zeta(2)		${\zeta}(\,2)$	$\frac{\pi^2}{6} = \sum_{n=1}^\infty\frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots$	Sum[n=1 to ∞] {1/n^2}	T	A013661	[1;1,1,1,4,2,4,7,1,4,2,3,4,10 1,2,1,1,1,15,...]	1826 to 1866	1.64493406684822643647241516664602519
1.73205 08075 68877 29352	Theodorus constant	100px	$\sqrt{3}$	$\sqrt[3]{3 \,\sqrt[3]{3 \, \sqrt[3]{3 \,\sqrt[3]{3 \,\sqrt[3]{3 \,\cdots}}}}}$	(3(3(3(3(3(3(3) ^1/3)^1/3)^1/3) ^1/3)^1/3)^1/3) ^1/3 ...	A	A002194	[1;1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,...] = [1;1,2,...]	-465 to -398	1.73205080756887729352744634150587237
1.75793 27566 18004 53270	Kasner number		${R}$	$\sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}}$	Fold[Sqrt[#1+#2] &,0,Reverse [Range[20]]]		A072449	[1;1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...]	1878 a 1955	1.75793275661800453270881963821813852
2.29558 71493 92638 07403	Universal parabolic constant	100px	${P}_{\,2}$	$\ln(1 + \sqrt2) + \sqrt2 \; = \; \operatorname{arcsinh}(1)+\sqrt{2}$	ln(1+sqrt 2)+sqrt 2	T	A103710	[2;3,2,1,1,1,1,3,3,1,1,4,2,3,2,7,1,6,1,8,7,2,1,...]		2.29558714939263807403429804918949038
1.78657 64593 65922 46345	Silverman constant		${\mathcal{S}_{_{m}}}$	$\sum_{n = 1}^\infty \frac {1}{\phi (n)\sigma_1(n)} = \underset{ p_n: \, {prime}}{ \prod_{n = 1}^\infty \left( 1 + \sum_{k = 1}^\infty \frac {1}{p_n^{2k} - p_n^{k-1}}\right)}$ ø() = Euler's totient function, σ1() = Divisor function.	Sum[n=1 to ∞] {1/[EulerPhi(n) DivisorSigma(1,n)]}		A093827	[1;1,3,1,2,5,1,65,11,2,1,2,13,1,4,1,1,1,2,5,4,...]		1.78657645936592246345859047554131575
2.59807 62113 53315 94029	Area of the regular hexagon with side equal to 1	80px	$\mathcal{A}_6$	$\frac{3 \sqrt{3}}{2}$	3 sqrt(3)/2	A	A104956	[2;1,1,2,20,2,1,1,4,1,1,2,20,2,1,1,4,1,1,2,20,...] [2;1,1,2,20,2,1,1,4]		2.59807621135331594029116951225880855
0.66131 70494 69622 33528	Feller-Tornier constant		${\mathcal{C}_{_{FT}}}$	$\underset{p_n: \, {prime}}{\frac{1}{2}\prod_{n = 1}^\infty \left(1-\frac{2}{p_n^2}\right){+}\frac{1}{2}} =\frac{3}{\pi^2}\prod_{n = 1}^\infty \left(1-\frac{1}{p_n^2-1}\right){+}\frac{1}{2}$	[prod[n=1 to ∞] {1-2/prime(n)^2}] /2 + 1/2	T ?	A065493	[0;1,1,1,20,9,1,2,5,1,2,3,2,3,38,8,1,16,2,2,...]	1932	0.66131704946962233528976584627411853
1.46099 84862 06318 35815	Baxter's Four-coloring constant	Mapamundi 100px Four-Coloring	$\mathcal{C}^2$	$\prod_{n = 1}^\infty \frac{(3n-1)^2}{(3n-2)(3n)} = \frac {3}{4\pi^2} \,\Gamma \left(\frac {1}{3}\right)^3$ Γ() = Gamma function	3×Gamma(1/3) ^3/(4 pi^2)		A224273	[1;2,5,1,10,8,1,12,3,1,5,3,5,8,2,1,23,1,2,161,...]	1970	1.46099848620631835815887311784605969
1.92756 19754 82925 30426	Tetranacci constant		$\mathcal{T}$	Positive root of $: \;\; x^4-x^3-x^2-x-1=0$	Root[x+x^-4-2=0]	A	A086088	[1;1,12,1,4,7,1,21,1,2,1,4,6,1,10,1,2,2,1,7,1,...]		1.92756197548292530426190586173662216
1.00743 47568 84279 37609	DeVicci's tesseract constant		${f_{(3,4)}}$	The largest cube that can pass through in an 4D hypercube. Positive root of $: \;\; 4x^4{-}28x^3{-}7x^2{+}16x{+}16=0$	Root[4*x^8-28*x^6 -7*x^4+16*x^2+16 =0]	A	A243309	[1;134,1,1,73,3,1,5,2,1,6,3,11,4,1,5,5,1,1,48,...]		1.00743475688427937609825359523109914
1.70521 11401 05367 76428	Niven's constant		${C}$	$1+\sum_{n = 2}^\infty \left(1-\frac{1}{\zeta(n)} \right)$	1+ Sum[n=2 to ∞] {1-(1/Zeta(n))}		A033150	[1;1,2,2,1,1,4,1,1,3,4,4,8,4,1,1,2,1,1,11,1,...]	1969	1.70521114010536776428855145343450816
0.60459 97880 78072 61686	Relationship among the area of an equilateral triangle and the inscribed circle.		$\frac{\pi}{3 \sqrt 3}$	$\sum_{n = 1}^\infty \frac{1}{n{2n \choose n}} = 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{5} + \frac{1}{7} - \frac{1}{8} + \cdots$ Dirichlet series	Sum[1/(n Binomial[2 n, n]) , {n, 1, ∞}]	T	A073010	[0;1,1,1,1,8,10,2,2,3,3,1,9,2,5,4,1,27,27,6,6,...]		0.60459978807807261686469275254738524
1.15470 05383 79251 52901	Hermite constant		$\gamma_{_{2}}$	$\frac{2}{\sqrt{3}} = \frac{1}{\cos \, (\frac{\pi}{6})}$	2/sqrt(3)	A	1+  A246724	[1;6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,...] [1;6,2]		1.15470053837925152901829756100391491
0.41245 40336 40107 59778	Prouhet–Thue–Morse constant	100px	$\tau$	$\sum_{n=0}^{\infty} \frac{t_n}{2^{n+1}}$    where ${t_n}$ is the Thue–Morse sequence  and Where $\tau(x) = \sum_{n=0}^{\infty} (-1)^{t_n} \, x^n = \prod_{n=0}^{\infty} ( 1 - x^{2^n} )$		T	A014571	[0;2,2,2,1,4,3,5,2,1,4,2,1,5,44,1,4,1,2,4,1,1,...]		0.41245403364010759778336136825845528
0.58057 75582 04892 40229	Pell constant		${\mathcal{P}_{_{Pell}}}$	$1- \prod_{n = 0}^\infty \left(1-\frac{1}{2^{2n+1}}\right)$	N[1-prod[n=0 to ∞] {1-1/(2^(2n+1)}]	T ?	A141848	[0;1,1,2,1,1,1,1,14,1,3,1,1,6,9,18,7,1,27,1,1,...]		0.58057755820489240229004389229702574
0.66274 34193 49181 58097	Laplace limit	100px	${\lambda}$	$\frac{ x \; e^\sqrt{x^2+1}}{\sqrt{x^2+1}+1} = 1$	(x e^sqrt(x^2+1)) /(sqrt(x^2+1)+1) = 1		A033259	[0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,1,1,2,1601,...]	1782 ~	0.66274341934918158097474209710925290
0.17150 04931 41536 06586	Hall-Montgomery Constant		${{\delta}_{_{0}}}$	$1 + \frac{\pi^2}{6} +2 \; \mathrm{Li}_2 \left(-\sqrt{e}\;\right) \quad \mathrm{Li}_2 \, \scriptstyle \text{= Dilogarithm integral}$	1 + Pi^2/6 + 2*PolyLog[2, -Sqrt[E]]		A143301	[0;5,1,4,1,10,1,1,11,18,1,2,19,14,1,51,1,2,1,...]		0.17150049314153606586043997155521210
1.55138 75245 48320 39226	Calabi triangle constant	100px	${C_{_{CR}}}$	${1 \over 3} + {(-23 + 3i \sqrt{237})^{\tfrac13} \over 3 \cdot 2^{\tfrac23}} + {11 \over 3 (2 (-23 + 3i \sqrt{237}))^{\tfrac13}}$	FindRoot[ 2x^3-2x^2-3x+2 ==0, {x, 1.5}, WorkingPrecision->40]	A	A046095	[1;1,1,4,2,1,2,1,5,2,1,3,1,1,390,1,1,2,11,6,2,...]	1946 ~	1.55138752454832039226195251026462381
1.22541 67024 65177 64512	Gamma(3/4)		$\Gamma(\tfrac34)$	$\left(-1+\frac{3}{4}\right)! = \left(-\frac{1}{4}\right)!$	(-1+3/4)!		A068465	[1;4,2,3,2,2,1,1,1,2,1,4,7,1,171,3,2,3,1,1,8,3,...]		1.22541670246517764512909830336289053
1.20205 69031 59594 28539	Apéry's constant	100px	$\zeta(3)$	$\sum_{n=1}^\infty\frac{1}{n^3} = \frac{1}{1^3}+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \cdots=$ $\frac{1}{2} \sum_{n=1}^\infty \frac{H_n}{n^2} = \frac{1}{2} \sum_{i=1}^\infty \sum_{j=1}^\infty \frac{1}{ij(i{+}j)}= \!\!\int \limits_0^1 \!\!\int \limits_0^1 \!\!\int \limits_0^1 \frac{\mathrm{d}x \mathrm{d}y \mathrm{d}z}{1 - xyz}$	Sum[n=1 to ∞] {1/n^3}	I	A010774	[1;4,1,18,1,1,1,4,1,9,9,2,1,1,1,2,7,1,1,7,11,...]	1979	1.20205690315959428539973816151144999
0.91596 55941 77219 01505	Catalan's constant		${C}$	$\int_0^1 \!\! \int_0^1 \!\! \frac{1}{1{+}x^2 y^2}\, dx \,dy = \! \sum_{n = 0}^\infty \! \frac{(-1)^n}{(2n{+}1)^2} \! = \! \frac{1}{1^2}{-}\frac{1}{3^2}{+}{\cdots}$	Sum[n=0 to ∞] {(-1)^n/(2n+1)^2}	T	A006752	[0;1,10,1,8,1,88,4,1,1,7,22,1,2,3,26,1,11,...]	1864	0.91596559417721901505460351493238411
0.78539 81633 97448 30961	Beta(1)		${\beta}(1)$	$\frac{\pi}{4} = \sum_{n = 0}^\infty \frac{(-1)^n}{2n+1} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots$	Sum[n=0 to ∞] {(-1)^n/(2n+1)}	T	A003881	[0; 1,3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...]	1805 to 1859	0.78539816339744830961566084581987572
0.00131 76411 54853 17810	Heath-Brown–Moroz constant		${C_{_{HBM}}}$	$\underset{p_n: \, {prime}}{\prod_{n = 1}^\infty \left(1-\frac{1}{p_n}\right)^7\left(1+\frac{7p_n+1}{p_n^2}\right)}$	N[prod[n=1 to ∞] {((1-1/prime(n))^7) *(1+(7*prime(n)+1) /(prime(n)^2))}]	T ?	A118228	[0;758,1,13,1,2,3,56,8,1,1,1,1,1,143,1,1,1,2,...]		0.00131764115485317810981735232251358
0.56755 51633 06957 82538	Module of Infinite Tetration of i		$|{}^\infty {i} |$	$\lim_{n \to \infty} \left | {}^n i \right | =\left | \lim_{n \to \infty} \underbrace{i^{i^{\cdot^{\cdot^{i}}}}}_n \right |$	Mod(i^i^i^...)		A212479	[0;1,1,3,4,1,58,12,1,51,1,4,12,1,1,2,2,3,...]		0.56755516330695782538461314419245334
0.78343 05107 12134 40705	Sophomore's dream1 J.Bernoulli	100px	${I}_{1}$	$\int_0^1 \! x^{x}\,dx = \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{n^n} = \frac{1}{1^1} - \frac{1}{2^2} + \frac{1}{3^3} - {\cdots}$	Sum[n=1 to ∞] {-(-1)^n /n^n}		A083648	[0;1,3,1,1,1,1,1,1,2,4,7,2,1,2,1,1,1,2,1,14,...]	1697	0.78343051071213440705926438652697546
1.29128 59970 62663 54040	Sophomore's dream2 J.Bernoulli	100px	${I}_{2}$	$\int_0^1 \! \frac{1}{x^x}\, dx = \sum_{n = 1}^\infty \frac{1}{n^n} = \frac{1}{1^1} + \frac{1}{2^2} + \frac{1}{3^3} + \frac{1}{4^4}+ \cdots$	Sum[n=1 to ∞] {1/(n^n)}		A073009	[1;3,2,3,4,3,1,2,1,1,6,7,2,5,3,1,2,1,8,1,2,4,...]	1697	1.29128599706266354040728259059560054
0.70523 01717 91800 96514	Primorial constant Sum of the product of inverse of primes		${P_\#}$	$\underset{ p_n: \, {prime}}{\sum_{n = 1}^\infty \frac{1}{p_n\#} = \frac{1}{2} + \frac{1}{6} + \frac{1}{30} + \frac{1}{210} + ... = \sum_{k = 1}^\infty \prod_{n = 1}^k \frac {1}{p_n}}$	Sum[k=1 to ∞] (prod[n=1 to k] {1/prime(n)})	I	A064648	[0;1,2,2,1,1,4,1,2,1,1,6,13,1,4,1,16,6,1,1,4,...]		0.70523017179180096514743168288824851
0.14758 36176 50433 27417	Plouffe's gamma constant	100px	${{C}}$	$\frac{1}{\pi} \arctan {\frac{1}{2}} = \frac{1}{\pi}\sum_{n=0}^\infty \frac {(-1)^n}{(2^{2n+1})(2n+1)}$ $= \frac{1}{\pi} \left( \frac {1}{2} - \frac {1}{3 \cdot 2^3} +\frac {1}{5 \cdot 2^5} -\frac {1}{7 \cdot 2^7} +\cdots \right)$	Arctan(1/2)/pi	T	A086203	[0;6,1,3,2,5,1,6,5,3,1,1,2,1,1,2,3,1,2,3,2,2,...]		0.14758361765043327417540107622474052
0.15915 49430 91895 33576	Plouffe's A constant		${A}$	$\frac{1}{2 \pi}$	1/(2 pi)	T	A086201	[0;6,3,1,1,7,2,146,3,6,1,1,2,7,5,5,1,4,1,2,42,...]		0.15915494309189533576888376337251436
0.29156 09040 30818 78013	Dimer constant 2D, Domino tiling	100px	${\frac{C}{\pi}}$ C=Catalan	$\int\limits_{-\pi}^{\pi} \frac{\cosh^{-1}\left(\frac{\sqrt{\cos(t)+3}}{\sqrt2}\right)}{4\pi}\,dt$	N[int[-pi to pi] {arccosh(sqrt( cos(t)+3)/sqrt(2)) /(4*Pi)dt}]		A143233	[0;3,2,3,16,8,10,3,1,1,2,1,3,1,2,13,1,1,4,1,5,...]		0.29156090403081878013838445646839491
0.49801 56681 18356 04271 0.15494 98283 01810 68512 i	Factorial(i)		${i}\,!$	$\Gamma (1+i) = i \, \Gamma (i) = \int\limits_0^\infty \frac{t^i}{e^t} \mathrm{d} t$	Integral_0^∞ t^i/e^t dt	C	A212877  A212878	[0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...] - [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i		0.49801566811835604271369111746219809 - 0.15494982830181068512495513048388 i
2.09455 14815 42326 59148	Wallis Constant	100px	$W$	$\sqrt[3]{\frac{45-\sqrt{1929}}{18}}+\sqrt[3]{\frac{45+\sqrt{1929}}{18}}$	(((45-sqrt(1929)) /18))^(1/3)+ (((45+sqrt(1929)) /18))^(1/3)	A	A007493	[2;10,1,1,2,1,3,1,1,12,3,5,1,1,2,1,6,1,11,4,...]	1616 to 1703	2.09455148154232659148238654057930296
0.72364 84022 98200 00940	Sarnak constant		${C_{sa} }$	$\prod_{p>2} \Big(1 - \frac{p+2}{p^3}\Big)$	N[prod[k=2 to ∞] {1-(prime(k)+2) /(prime(k)^3)}]	T ?	A065476	[0;1,2,1,1,1,1,1,1,1,4,4,1,1,1,1,1,1,1,8,2,1,1,...]		0.72364840229820000940884914980912759
0.63212 05588 28557 67840	Time constant	100px	${\tau}$	$\lim_{n \to \infty} 1-\frac {!n}{n!}=\lim_{n \to \infty} P(n)= \int_{0}^{1}e^{-x}dx = 1{-}\frac{1}{e} =$ $\sum \limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{n!} = \frac{1}{1!}{-}\frac{1}{2!}{+}\frac{1}{3!}{-}\frac{1}{4!}{+}\frac{1}{5!}{-}\frac{1}{6!}{+}\cdots$	lim_(n->∞) (1- !n/n!) !n=subfactorial	T	A068996	[0;1,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...] = [0;1,1,1,2n], n∈ℕ		0.63212055882855767840447622983853913
1.04633 50667 70503 18098	Minkowski-Siegel mass constant		$F_1$	$\prod_{n=1}^{\infty} \frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n \sqrt[12]{1+\tfrac1{n}}}$	N[prod[n=1 to ∞] n! /(sqrt(2*Pi*n) *(n/e)^n *(1+1/n) ^(1/12))]		A213080	[1;21,1,1,2,1,1,4,2,1,5,7,2,1,20,1,1,1134,3,..]	1867 1885 1935	1.04633506677050318098095065697776037
5.24411 51085 84239 62092	Lemniscate Constant		$2\varpi$	$\frac{[\Gamma(\tfrac14)]^2}{\sqrt{2 \pi}} = 4\int^1_0 \frac{dx}{\sqrt{(1-x^2)(2-x^2)}}$	Gamma[ 1/4 ]^2 /Sqrt[ 2 Pi ]		A064853	[5;4,10,2,1,2,3,29,4,1,2,1,2,1,2,1,4,9,1,4,1,2,...]	1718	5.24411510858423962092967917978223883
0.66170 71822 67176 23515	Robbins constant		$\Delta(3)$	$\frac{4 \! + \! 17\sqrt2 \! -6 \sqrt3 \! -7\pi}{105} \! + \! \frac{\ln(1 \! + \! \sqrt2)}{5} \! + \! \frac{2\ln(2 \! + \! \sqrt3)}{5}$	(4+17*2^(1/2)-6 *3^(1/2)+21*ln(1+ 2^(1/2))+42*ln(2+ 3^(1/2))-7*Pi)/105		A073012	[0;1,1,1,21,1,2,1,4,10,1,2,2,1,3,11,1,331,1,4,...]	1978	0.66170718226717623515583113324841358
1.30357 72690 34296 39125	Conway constant	100px	${\lambda}$	$\begin{smallmatrix} x^{71}\quad\ -x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\ -x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\ +x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\ -12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\ -10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}\quad\ -7x^{21}+9x^{20}\\ +3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\ +5x^{9}+x^{7}\quad\ -7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0 \quad\quad\quad \end{smallmatrix}$		A	A014715	[1;3,3,2,2,54,5,2,1,16,1,30,1,1,1,2,2,1,14,1,...]	1987	1.30357726903429639125709911215255189
1.18656 91104 15625 45282	Khinchin–Lévy constant		${\beta}$	$\frac {\pi^2}{12\,\ln 2}$	pi^2 /(12 ln 2)		A100199	[1;5,2,1,3,1,1,28,18,16,3,2,6,2,6,1,1,5,5,9,...]	1935	1.18656911041562545282172297594723712
0.83564 88482 64721 05333	Baker constant	100px	$\beta_3$	$\int^1_0 \frac{{\mathrm{d} t}}{1 + t^3}=\sum_{n = 0}^\infty \frac{(-1)^n}{3n+1}= \frac{1}{3}\left(\ln 2+\frac{\pi}{\sqrt{3}}\right)$	Sum[n=0 to ∞] {((-1)^(n))/(3n+1)}		A113476	[0;1,5,11,1,4,1,6,1,4,1,1,1,2,1,3,2,2,2,2,1,3,...]		0.83564884826472105333710345970011076
23.10344 79094 20541 6160	Kempner Serie(0)		${K_0}$	$1{+}\frac12{+}\frac13{+}\cdots{+}\frac19{+}\frac1{11}{+}\cdots{+}\frac1{19}{+}\frac1{21}{+}\cdots$ ${+}\frac1{99}{+}\frac1{111}{+}\cdots{+}\frac1{119}{+}\frac1{121}{+}\cdots$ (Excluding all denominators containing 0.)	1+1/2+1/3+1/4+1/5 +1/6+1/7+1/8+1/9 +1/11+1/12+1/13 +1/14+1/15+...		A082839	[23;9,1,2,3244,1,1,5,1,2,2,8,3,1,1,6,1,84,1,...]		23.1034479094205416160340540433255981
0.98943 12738 31146 95174	Lebesgue constant	100px	${C_1}$	$\lim_{n\to\infty}\!\! \left(\!{L_n{-}\frac{4}{\pi^2}\ln(2n{+}1)}\!\!\right)\!{=} \frac{4}{\pi^2}\!\left({\sum_{k=1}^\infty \!\frac{2\ln k}{4k^2{-}1}} {-}\frac{\Gamma'(\tfrac12)}{\Gamma(\tfrac12)}\!\!\right)$	4/pi^2*[(2 Sum[k=1 to ∞] {ln(k)/(4*k^2-1)}) -poligamma(1/2)]		A243277	[0;1,93,1,1,1,1,1,1,1,7,1,12,2,15,1,2,7,2,1,5,...]	?	0.98943127383114695174164880901886671
0.19452 80494 65325 11361	2nd du Bois-Reymond constant		${C_2}$	$\frac{e^2-7}{2} = \int_0^\infty \left|{\frac{d}{dt}\left(\frac{\sin t}{t}\right)^n}\right|\,dt-1$	(e^2-7)/2	T	A062546	[0;5,7,9,11,13,15,17,19,21,23,25,27,29,31,...] = [0;2p+3], p∈ℕ		0.19452804946532511361521373028750390
0.78853 05659 11508 96106	Lüroth constant	35px	$C_L$	$\sum_{n = 2}^\infty \frac{\ln\left(\frac{n}{n-1}\right)}{n}$	Sum[n=2 to ∞] log(n/(n-1))/n		A085361	[0;1,3,1,2,1,2,4,1,127,1,2,2,1,3,8,1,1,2,1,16,...]		0.78853056591150896106027632216944432
1.18745 23511 26501 05459	Foias constant α		$F_\alpha$	$x_{n+1} = \left( 1 + \frac{1}{x_n} \right)^n\text{ for }n=1,2,3,\ldots$ Foias constant is the unique real number such that if x1 = α then the sequence diverges to ∞. When x1 = α, $\, \lim_{n\to\infty} x_n \tfrac{\log n}{n} = 1$			A085848	[1;5,2,1,81,3,2,2,1,1,1,1,1,6,1,1,3,1,1,4,3,2,...]	2000	1.18745235112650105459548015839651935
2.29316 62874 11861 03150	Foias constant β	100px	$F_\beta$	$x^{x+1} = (x+1)^x$	x^(x+1) = (x+1)^x		A085846	[2;3,2,2,3,4,2,3,2,130,1,1,1,1,1,6,3,2,1,15,1,...]	2000	2.29316628741186103150802829125080586
0.82246 70334 24113 21823	Nielsen-Ramanujan constant		$\frac{{\zeta}(2)}{2}$	$\frac{\pi^2}{12} = \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^2} = \frac{1}{1^2} {-} \frac{1}{2^2} {+} \frac{1}{3^2} {-} \frac{1}{4^2} {+} \frac{1}{5^2} {-} \cdots$	Sum[n=1 to ∞] {((-1)^(n+1))/n^2}	T	A072691	[0;1,4,1,1,1,2,1,1,1,1,3,2,2,4,1,1,1,1,1,1,4...]	1909	0.82246703342411321823620758332301259
0.69314 71805 59945 30941	Natural logarithm of 2		$Ln(2)$	$\sum_{n=1}^\infty \frac{1}{n 2^n} = \sum_{n=1}^\infty \frac{({-}1)^{n+1}}{n} = \frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+{\cdots}$	Sum[n=1 to ∞] {(-1)^(n+1)/n}	T	A002162	[0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,1,1,1,2,1,1,...]	1550 to 1617	0.69314718055994530941723212145817657
0.47494 93799 87920 65033	Weierstrass constant		$\sigma(\tfrac12)$	$\frac{e^{\frac{\pi}{8}}\sqrt{\pi}}{4 \cdot 2^{3/4} {(\frac {1}{4}!)^2}}$	(E^(Pi/8) Sqrt[Pi]) /(4 2^(3/4) (1/4)!^2)		A094692	[0;2,9,2,11,1,6,1,4,6,3,19,9,217,1,2,4,8,6...]	1872 ?	0.47494937998792065033250463632798297
0.57721 56649 01532 86060	Euler–Mascheroni constant	100px	${\gamma}$	$\sum_{n=1}^\infty \sum_{k=0}^\infty \frac{(-1)^k}{2^n+k} = \sum_{n=1}^\infty \left(\frac{1}{n} -\ln \left(1+\frac{1}{n}\right)\right)$ $= \int_{0}^{1} -\ln \left(\ln \frac{1}{x}\right)\, dx = -\Gamma'(1) = -\Psi(1)$	sum[n=1 to ∞] |sum[k=0 to ∞] {((-1)^k)/(2^n+k)}		A001620	[0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,...]	1735	0.57721566490153286060651209008240243
1.38135 64445 18497 79337	Beta, Kneser-Mahler polynomial constant		$\beta$	$e^{^{\textstyle{\frac{2}{\pi}} \displaystyle{\int_0^{\frac{\pi}{3}}} \textstyle{t \tan t\ dt}}} = e^{^{\displaystyle{\,\int_{\frac{-1}{3}}^{\frac{1}{3}}} \textstyle{\,\ln \lfloor 1+e^{2 \pi i t}} \rfloor dt}}$	e^((PolyGamma(1,4/3) - PolyGamma(1,2/3) +9)/(4*sqrt(3)*Pi))		A242710	[1;2,1,1,1,1,1,4,1,139,2,1,3,5,16,2,1,1,7,2,1,...]	1963	1.38135644451849779337146695685062412
1.35845 62741 82988 43520	Golden Spiral		$c$	$\varphi ^ \frac{2}{\pi} = \left(\frac{1 + \sqrt{5}}{2}\right)^{\frac{2}{\pi}}$	GoldenRatio^(2/pi)		A212224	[1;2,1,3,1,3,10,8,1,1,8,1,15,6,1,3,1,1,2,3,1,1,...]		1.35845627418298843520618060050187945
0.57595 99688 92945 43964	Stephens constant		$C_S$	$\prod_{n = 1}^\infty \left(1 - \frac{p}{p^3-1}\right)$	Prod[n=1 to ∞] {1-hprime(n) /(hprime(n)^3-1)}	T ?	A065478	[0;1,1,2,1,3,1,3,1,2,1,77,2,1,1,10,2,1,1,1,7,...]	?	0.57595996889294543964316337549249669
0.73908 51332 15160 64165	Dottie number	100px	$d$	$\lim_{x\to \infty} \cos^{[x]}(c) = \lim_{x\to \infty} \underbrace{\cos(\cos(\cos(\cdots(\cos(c)))))}_x$	cos(c)=c	T	A003957	[0;1,2,1,4,1,40,1,9,4,2,1,15,2,12,1,21,1,17,...]	?	0.73908513321516064165531208767387340
0.67823 44919 17391 97803	Taniguchi constant		$C_T$	$\prod_{n = 1}^\infty \left(1 - \frac{3}{{p_n}^3}+\frac{2}{{p_n}^4}+\frac{1}{{p_n}^5}-\frac{1}{{p_n}^6}\right)$ $\scriptstyle p_{n}= \, \text{prime}$	Prod[n=1 to ∞] {1 -3/ithprime(n)^3 +2/ithprime(n)^4 +1/ithprime(n)^5 -1/ithprime(n)^6}	T ?	A175639	[0;1,2,9,3,1,2,9,11,1,13,2,15,1,1,1,2,4,1,1,1,...]	?	0.67823449191739197803553827948289481
1.85407 46773 01371 91843	Gauss' Lemniscate constant	100px	$L \text{/}\sqrt{2}$	$\int\limits_0^\infty \frac{{\mathrm{d} x}}{\sqrt{1 + x^4}} = \frac {1}{4\sqrt{\pi}} \,\Gamma \left(\frac {1}{4}\right)^2 = \frac{4 \left(\frac {1}{4}!\right)^2} {\sqrt{\pi}}$ $\scriptstyle \Gamma() \text{= Gamma function}$	pi^(3/2)/(2 Gamma(3/4)^2)		A093341	[1;1,5,1,5,1,3,1,6,2,1,4,16,3,112,2,1,1,18,1,...]		1.85407467730137191843385034719526005
1.75874 36279 51184 82469	Infinite product constant, with Alladi-Grinstead		$Pr_1$	$\prod_{n = 2}^\infty \Big(1 + \frac{1}{n}\Big)^\frac{1}{n}$	Prod[n=2 to inf] {(1+1/n)^(1/n)}		A242623	[1;1,3,6,1,8,1,4,3,1,4,1,1,1,6,5,2,40,1,387,2,...]	1977	1.75874362795118482469989684865589317
1.86002 50792 21190 30718	Spiral of Theodorus		$\partial$	$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3} + \sqrt{n}} = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n} (n+1)}$	Sum[n=1 to ∞] {1/(n^(3/2) +n^(1/2))}		A226317	[1;1,6,6,1,15,11,5,1,1,1,1,5,3,3,3,2,1,1,2,19,...]	-460 to -399	1.86002507922119030718069591571714332
2.79128 78474 77920 00329	Nested radical S5		$S_{5}$	$\displaystyle \frac{\sqrt{21}+1}{2} = \scriptstyle \, \sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}}}}\;$ $= 1+ \, \scriptstyle \sqrt{5-\sqrt{5-\sqrt{5-\sqrt{5-\sqrt{5-\cdots}}}}}\;$	(sqrt(21)+1)/2	A	A222134	[2;1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,...] [2;1,3]	?	2.79128784747792000329402359686400424
0.70710 67811 86547 52440 +0.70710 67811 86547 524 i	Square root of i		$\sqrt{i}$	$\sqrt[4]{-1} = \frac{1+i}{\sqrt{2}} = e^ \frac{i\pi}{4} = \cos\left (\frac{\pi}{4} \right ) + i\sin\left ( \frac{\pi}{4} \right )$	(1+i)/(sqrt 2)	C A	A010503	[0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,..] = [0;1,2,...] [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,..] i = [0;1,2,...] i	?	0.70710678118654752440084436210484903 + 0.70710678118654752440084436210484 i
0.80939 40205 40639 13071	Alladi–Grinstead constant		${\mathcal{A}_{AG}}$	$e^{-1+\sum \limits_{k=2}^\infty \sum \limits_{n=1}^\infty \frac{1}{n k^{n+1}}} = e^{-1-\sum \limits_{k=2}^\infty \frac{1}{k} \ln \left( 1-\frac{1}{k}\right)}$	e^{(sum[k=2 to ∞] |sum[n=1 to ∞] {1/(n k^(n+1))})-1}		A085291	[0;1,4,4,17,4,3,2,5,3,1,1,1,1,6,1,1,2,1,22,...]	1977	0.80939402054063913071793188059409131
2.58498 17595 79253 21706	Sierpiński's constant		${K}$	$\pi\left(2\gamma+\ln\frac{4\pi^3}{\Gamma(\tfrac{1}{4})^4}\right) = \pi (2 \gamma + 4 \ln\Gamma(\tfrac{3}{4}) - \ln\pi)$ $= \pi \left(2 \ln 2+3 \ln \pi + 2 \gamma - 4 \ln \Gamma (\tfrac{1}{4})\right)$	-Pi Log[Pi]+2 Pi EulerGamma +4 Pi Log [Gamma[3/4]]		A062089	[2;1,1,2,2,3,1,3,1,9,2,8,4,1,13,3,1,15,18,1,...]	1907	2.58498175957925321706589358738317116
1.73245 47146 00633 47358	Reciprocal of the Euler–Mascheroni constant		$\frac {1}{\gamma}$	$\left(\int_{0}^{1} -\log \left(\log \frac{1}{x}\right)\, dx\right)^{-1} = \sum_{n=1}^\infty (-1)^n (-1+\gamma)^n$	1/Integrate_ {x=0 to 1} -log(log(1/x))		A098907	[1;1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,...]		1.73245471460063347358302531586082968
1.43599 11241 76917 43235	Lebesgue constant (interpolation)		${L_1}$	$\prod_{\begin{smallmatrix}i=0\\ j\neq i\end{smallmatrix}}^{n} \frac{x-x_i}{x_j-x_i} = \frac {1}{\pi} \int_0^{\pi} \frac {\lfloor \sin{\frac{3 t}{2}}\rfloor}{\sin{\frac{t}{2}}}\, dt = \frac {1}{3} + \frac {2 \sqrt{3}}{\pi}$	1/3 + 2*sqrt(3)/pi	T	A226654	[1;2,3,2,2,6,1,1,1,1,4,1,7,1,1,1,2,1,3,1,2,1,1,...]	1902 ~	1.43599112417691743235598632995927221
3.24697 96037 17467 06105	Silver root Tutte–Beraha constant		$\varsigma$	$2+2 \cos \frac {2\pi} 7 = \textstyle 2+\frac{2+\sqrt[3]{7 + 7 \sqrt[3]{7 + 7 \sqrt[3]{\, 7 + \cdots}}}}{1+\sqrt[3]{7 + 7 \sqrt[3]{7 + 7 \sqrt[3]{\, 7 + \cdots}}}}$	2+2 cos(2Pi/7)	A	A116425	[3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...]		3.24697960371746706105000976800847962
1.94359 64368 20759 20505	Euler Totient constant		$ET$	$\underset {p \text{= primes}} {\prod_{p} \Big(1 + \frac{1}{p(p-1)}\Big)} = \frac {\zeta(2)\zeta(3)}{\zeta(6)}=\frac {315 \zeta(3)}{2\pi^4}$	zeta(2)*zeta(3) /zeta(6)		A082695	[1;1,16,1,2,1,2,3,1,1,3,2,1,8,1,1,1,1,1,1,1,32,...]	1750	1.94359643682075920505707036257476343
1.49534 87812 21220 54191	Fourth root of five		$\sqrt[4]{5}$	$\sqrt[5]{5 \,\sqrt[5]{5 \, \sqrt[5]{5 \,\sqrt[5]{5 \,\sqrt[5]{5 \,\cdots}}}}}$	(5(5(5(5(5(5(5) ^1/5)^1/5)^1/5) ^1/5)^1/5)^1/5) ^1/5 ...	A	A011003	[1;2,53,4,96,2,1,6,2,2,2,6,1,4,1,49,17,2,3,2,...]		1.49534878122122054191189899414091339
0.87228 40410 65627 97617	Area of Ford circle		$A_{CF}$	$\sum_{q\ge 1} \sum_{ (p, q)=1 \atop 1 \le p < q }\pi \left( \frac{1}{2 q^2} \right)^2 \underset {\zeta() \text{= Riemann Zeta Function}} {= \frac{\pi}{4} \frac{\zeta(3)}{\zeta(4)} = \frac{45}{2} \frac{\zeta(3)}{\pi^3}}$	pi Zeta(3) /(4 Zeta(4))			[0;1,6,1,4,1,7,5,36,3,29,1,1,10,3,2,8,1,1,1,3,...]		0.87228404106562797617519753217122587
1.08232 32337 11138 19151	Zeta(4)		$\zeta(4)$	$\frac{\pi^4}{90} = \sum_{n=1}^\infty\frac{{1}}{n^4} = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \frac{1}{5^4} + ...$	Sum[n=1 to ∞] {1/n^4}	T	A013662	[1;12,6,1,3,1,4,183,1,1,2,1,3,1,1,5,4,2,7,23,...]	?	1.08232323371113819151600369654116790
1.56155 28128 08830 27491	Triangular root of 2.	100px	${R_2}$	$\frac{\sqrt{17}-1}{2} = \,\scriptstyle \sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\cdots}}}}}} \,\, -1$ $= \,\scriptstyle \sqrt{4-\sqrt{4-\sqrt{4-\sqrt{4-\sqrt{4-\sqrt{4-\cdots}}}}}} \textstyle$	(sqrt(17)-1)/2	A	A222133	[1;1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,...] [1;1,1,3]		1.56155281280883027491070492798703851
9.86960 44010 89358 61883	Pi Squared		${\pi} ^2$	$6\, \zeta(2) = 6 \sum_{n=1}^\infty \frac{1}{n^2} = \frac{6}{1^2} + \frac{6}{2^2} + \frac{6}{3^2} + \frac{6}{4^2}+ \cdots$	6 Sum[n=1 to ∞] {1/n^2}	T	A002388	[9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,1,3,...]		9.86960440108935861883449099987615114
1.32471 79572 44746 02596	Plastic number	100px	${\rho}$	$\sqrt[3]{1 + \! \sqrt[3]{1 + \! \sqrt[3]{1 + \cdots}}} = \textstyle \sqrt[3]{\frac{1}{2}+ \! \sqrt{\frac{23}{108}}}+ \! \sqrt[3]{\frac{1}{2}- \! \sqrt{\frac{23}{108}}}$	(1+(1+(1+(1+(1+(1) ^(1/3))^(1/3))^(1/3)) ^(1/3))^(1/3))^(1/3)	A	A060006	[1;3,12,1,1,3,2,3,2,4,2,141,80,2,5,1,2,8,2,...]	1929	1.32471795724474602596090885447809734
2.37313 82208 31250 90564	Lévy 2 constant		$2\,ln\,\gamma$	$\frac{\pi^2}{6ln(2)}$	Pi^(2)/(6*ln(2))	T	A174606	[2;2,1,2,8,57,9,32,1,1,2,1,2,1,2,1,2,1,3,2,...]	1936	2.37313822083125090564344595189447424
0.85073 61882 01867 26036	Regular paperfolding sequence	100px	${P_f}$	$\sum_{n=0}^{\infty} \frac {8^{2^n}}{2^{2^{n+2}}-1} = \sum_{n=0}^{\infty} \cfrac {\tfrac {1}{2^{2^n}}} {1-\tfrac{1}{2^{2^{n+2}}}}$	N[Sum[n=0 to ∞] {8^2^n/(2^2^ (n+2)-1)},37]		A143347	[0;1,5,1,2,3,21,1,4,107,7,5,2,1,2,1,1,2,1,6,...]		0.85073618820186726036779776053206660
1.15636 26843 32269 71685	Cubic recurrence constant {{.}}		${\sigma_3}$	$\prod_{n=1}^\infty n^{{3}^{-n}} = \sqrt[3] {1 \sqrt[3] {2 \sqrt[3]{3 \cdots}}} = 1^{1/3} \; 2^{1/9} \; 3^{1/27} \cdots$	prod[n=1 to ∞] {n ^(1/3)^n}		A123852	[1;6,2,1,1,8,13,1,3,2,2,6,2,1,2,1,1,1,10,33,...]		1.15636268433226971685337032288736935
1.26185 95071 42914 87419	Fractal dimension of the Koch snowflake	100px	${C_k}$	$\frac{\log 4}{\log 3}$	log(4)/log(3)	T	A100831	[1;3,1,4,1,1,11,1,46,1,5,112,1,1,1,1,1,3,1,7,...]		1.26185950714291487419905422868552171
6.58088 59910 17920 97085	Froda constant		$2^{\,e}$	$2^e$	2^e			[6;1,1,2,1,1,2,3,1,14,11,4,3,1,1,7,5,5,2,7,...]		6.58088599101792097085154240388648649
0.26149 72128 47642 78375	Meissel-Mertens constant	100px	${M}$	$\lim_{n \rightarrow \infty } \!\! \left( \sum_{p \leq n} \frac{1}{p} \! - \ln(\ln(n))\! \right) \!\! = \underset{\!\!\!\! \gamma: \, \text{Euler constant} ,\,\, p: \, \text{prime}}{\! \gamma \! + \!\! \sum_{p} \!\left( \! \ln \! \left( \! 1 \! - \! \frac{1}{p} \! \right) \!\! + \! \frac{1}{p} \! \right)}$	gamma+ Sum[n=1 to ∞] {ln(1-1/prime(n)) +1/prime(n)}	T ?	A077761	[0;3,1,4,1,2,5,2,1,1,1,1,13,4,2,4,2,1,33,296,...]	1866 & 1873	0.26149721284764278375542683860869585
4.81047 73809 65351 65547	John constant		$\gamma$	$\sqrt[i]{i} = i^{-i} = (i^i)^{-1} = (((i)^i)^i)^i = e^{\frac{\pi}{2}} = \sqrt{\sum_{n=0}^\infty \frac{\pi^{n}}{n!}}$	e^(π/2)	T	A042972	[4;1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,3,2,...]		4.81047738096535165547303566670383313
-0.5 ± 0.86602 54037 84438 64676 i	Cube Root of 1		$\sqrt[3]{1}$	$\begin{cases} \ \ 1 \\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i \\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i. \end{cases}$	1, E^(2i pi/3), E^(-2i pi/3)	C A	A010527	- [0,5] ± [0;1,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,...] i - [0,5] ± [0; 1, 6, 2] i		- 0.5 ± 0.8660254037844386467637231707529 i
0.11000 10000 00000 00000 0001	Liouville number		$\text{£}_{Li}$	$\sum_{n=1}^\infty \frac{1}{10^{n!}} = \frac {1}{10^{1!}} + \frac{1}{10^{2!}} + \frac{1}{10^{3!}} + \frac{1}{10^{4!}} + \cdots$	Sum[n=1 to ∞] {10^(-n!)}	T	A012245	[1;9,1,999,10,9999999999999,1,9,999,1,9]		0.11000100000000000000000100...
0.06598 80358 45312 53707	Lower limit of Tetration		${e}^{-e}$	$\left(\frac {1}{e}\right)^e$	1/(e^e)		A073230	[0;15,6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,...]		0.06598803584531253707679018759684642
1.83928 67552 14161 13255	Tribonacci constant		${\phi_{}}_3$	$\textstyle \frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3} = \scriptstyle \, 1+ \left(\sqrt[3]{\tfrac12 + \sqrt[3]{\tfrac12 + \sqrt[3]{\tfrac12 + ...}}}\right)^{-1}$	(1/3)*(1+(19+3 *sqrt(33))^(1/3) +(19-3 *sqrt(33))^(1/3))	A	A058265	[1;1,5,4,2,305,1,8,2,1,4,6,14,3,1,13,5,1,7,...]		1.83928675521416113255185256465328660
0.36651 29205 81664 32701	Median of the Gumbel distribution	100px	${ll_2}$	$-\ln(\ln(2))$	-ln(ln(2))		A074785	[0;2,1,2,1,2,6,1,6,6,2,2,2,1,12,1,8,1,1,3,1,...]		0.36651292058166432701243915823266947
36.46215 96072 07911 7709	Pi^pi		$\pi ^\pi$	$\pi ^\pi$	pi^pi		A073233	[36;2,6,9,2,1,2,5,1,1,6,2,1,291,1,38,50,1,2,...]		36.4621596072079117709908260226921236
0.53964 54911 90413 18711	Ioachimescu constant		$2+\zeta(\tfrac12)$	${2{-}(1{+}\sqrt{2})\sum_{n=1}^\infty \frac{(-1)^{n+1}}{\sqrt{n}}} = \gamma + \sum_{n=1}^\infty \frac{(-1)^{2n} \; \gamma_n}{2^n n!}$	γ + N[ sum[n=1 to ∞] {((-1)^(2n) gamma_n) /(2^n n!)}]		2-  A059750	[0;1,1,5,1,4,6,1,1,2,6,1,1,2,1,1,1,37,3,2,1,...]		0.53964549119041318711050084748470198
15.15426 22414 79264 1897	Exponential reiterated constant	100px	$e^e$	$\sum_{n=0}^\infty \frac{e^n}{n!} = \lim_{n \to \infty} \left(\frac {1+n}{n} \right)^{n^{-n}(1+n)^{1+n}}$	Sum[n=0 to ∞] {(e^n)/n!}		A073226	[15;6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,6,7,...]		15.1542622414792641897604302726299119
0.64624 54398 94813 30426	Masser–Gramain constant		${C}$	$\gamma {\beta}(1) \! + \! {\beta}'(1) \! = \pi \! \left(-\!\ln \Gamma(\tfrac14)+\tfrac34 \pi+\tfrac12 \ln 2+\tfrac12 \gamma \right)$ $= \pi \! \left(-\!\ln (\tfrac14 !)+\tfrac34 \ln \pi -\tfrac32 \ln 2+\tfrac12 \, \gamma \right)$ $\scriptstyle \gamma = \text{Euler–Mascheroni constant}= 0.5772156649\ldots$ $\scriptstyle \beta() = \text{Beta function} , \quad \scriptstyle \Gamma() = \text{Gamma function}$	Pi/4*(2*Gamma + 2*Log[2] + 3*Log[Pi]- 4 Log[Gamma[1/4]])		A086057	[0;1,1,1,4,1,3,2,3,9,1,33,1,4,3,3,5,3,1,3,4,...]		0.64624543989481330426647339684579279
1.11072 07345 39591 56175	The ratio of a square and circle circumscribed	100px	$\frac{\pi}{2\sqrt 2}$	$\sum_{n = 1}^\infty \frac{({-}1)^{\lfloor \frac{n-1}{2}\rfloor}}{2n+1} = \frac{1}{1} + \frac{1}{3} - \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - {\cdots}$	sum[n=1 to ∞] {(-1)^(floor( (n-1)/2)) /(2n-1)}	T	A093954	[1;9,31,1,1,17,2,3,3,2,3,1,1,2,2,1,4,9,1,3,...]		1.11072073453959156175397024751517342
1.45607 49485 82689 67139	Backhouse's constant		${B}$	$\lim_{k \to \infty}\left | \frac{q_{k+1}}{q_k} \right \vert \quad \scriptstyle \text {where:} \displaystyle \;\; Q(x)=\frac{1}{P(x)}= \! \sum_{k=1}^\infty q_k x^k$ $P(x) = \sum_{k=1}^\infty \underset{p_k\text{ prime}}{p_k x^k} = 1+2x+3x^2+5x^3+\cdots$	1/( FindRoot[0 == 1 + Sum[x^n Prime[n], {n, 10000}], {x, {1}})		A072508	[1;2,5,5,4,1,1,18,1,1,1,1,1,2,13,3,1,2,4,16,...]	1995	1.45607494858268967139959535111654355
1.85193 70519 82466 17036	Gibbs constant		${Si(\pi)}$ Sin integral	$\int_0^{\pi} \frac {\sin t}{t}\, dt = \sum \limits_{n=1}^\infty (-1)^{n-1} \frac{\pi^{2n-1}}{(2n-1)(2n-1)!}$ $= \pi- \frac{\pi^3}{3\cdot3!} + \frac{\pi^5}{5\cdot5!} - \frac{\pi^7}{7\cdot7!} + \cdots$	SinIntegral[Pi]		A036792	[1;1,5,1,3,15,1,5,3,2,7,2,1,62,1,3,110,1,39,...]		1.85193705198246617036105337015799136
0.23571 11317 19232 93137	Copeland–Erdős constant		${\mathcal{C}_{CE}}$	$\sum _{n=1}^\infty \frac{p_n} {10^{n + \sum \limits_{k=1}^n \lfloor \log_{10}{p_k} \rfloor }}$	sum[n=1 to ∞] {prime(n) /(n+(10^ sum[k=1 to n]{floor (log_10 prime(k))}))}	I	A033308	[0;4,4,8,16,18,5,1,1,1,1,7,1,1,6,2,9,58,1,3,...]		0.23571113171923293137414347535961677
1.52362 70862 02492 10627	Fractal dimension of the boundary of the dragon curve		${C_d}$	$\frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)} {\log(2)}$	(log((1+(73-6 sqrt(87))^1/3+ (73+6 sqrt(87))^1/3)/3))/ log(2)))	T		[1;1,1,10,12,2,1,149,1,1,1,3,11,1,3,17,4,1,...]		1.52362708620249210627768393595421662
1.78221 39781 91369 11177	Grothendieck constant		${K_{R}}$	$\frac {\pi}{2 \log(1+\sqrt{2})}$	pi/(2 log(1+sqrt(2)))		A088367	[1;1,3,1,1,2,4,2,1,1,17,1,12,4,3,5,10,1,1,3,...]		1.78221397819136911177441345297254934
1.58496 25007 21156 18145	Hausdorff dimension, Sierpinski triangle	100px	${log_2 3}$	$\frac {\log 3}{\log 2} = \frac{\sum_{n=0}^\infty \frac{1}{2^{2n+1}(2n+1)}}{\sum_{n=0}^\infty \frac{1}{3^{2n+1}(2n+1)}} = \frac{\frac{1}{2}+\frac{1}{24}+\frac{1}{160}+\cdots}{\frac{1}{3}+\frac{1}{81}+\frac{1}{1215}+\cdots}$	( Sum[n=0 to ∞] {1/ (2^(2n+1) (2n+1))})/ (Sum[n=0 to ∞] {1/ (3^(2n+1) (2n+1))})	T	A020857	[1;1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...]		1.58496250072115618145373894394781651
1.30637 78838 63080 69046	Mills' constant		${\theta}$	$\lfloor \theta^{3^{n}} \rfloor$ primes	Nest[ NextPrime[#^3] &, 2, 7]^(1/3^8)		A051021	[1;3,3,1,3,1,2,1,2,1,4,2,35,21,1,4,4,1,1,3,2,...]	1947	1.30637788386308069046861449260260571
2.02988 32128 19307 25004	Figure eight knot hyperbolic volume		${V_{8}}$	$2 \sqrt{3}\, \sum_{n=1}^\infty \frac{1}{n {2n \choose n}} \sum_{k=n}^{2n-1} \frac{1}{k} = 6 \int \limits_{0}^{\pi / 3} \log \left( \frac{1}{2 \sin t} \right) \, dt =$ $\scriptstyle \frac{\sqrt{3}}{{9}}\, \sum \limits_{n=0}^\infty \frac{(-1)^n}{27^n}\,\left\{\! \frac{{18}}{(6n+1)^2} - \frac{{18}}{(6n+2)^2} - \frac{{24}}{(6n+3)^2} - \frac{{6}}{(6n+4)^2} + \frac{{2}}{(6n+5)^2}\!\right\}$	6 integral[0 to pi/3] {log(1/(2 sin (n)))}		A091518	[2;33,2,6,2,1,2,2,5,1,1,7,1,1,1,113,1,4,5,1,...]		2.02988321281930725004240510854904057
262 53741 26407 68743 .99999 99999 99250 073	Hermite–Ramanujan constant		${R}$	$e^{\pi\sqrt{163}}$	e^(π sqrt(163))	T	A060295	[262537412640768743;1,1333462407511,1,8,1,1,5,...]	1859	262537412640768743.999999999999250073
1.74540 56624 07346 86349	Khinchin harmonic mean	100px	${K_{-1}}$	$\frac {\log 2} {\sum \limits_{n=1}^\infty \frac {1}{n} \log\bigl(1{+}\frac{1}{n(n+2)}\bigr)} = \lim_{n \to \infty} \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}$ a1 ... an are elements of a continued fraction [a0; a1, a2, ..., an]	(log 2)/ (sum[n=1 to ∞] {1/n log(1+ 1/(n(n+2))}		A087491	[1;1,2,1,12,1,5,1,5,13,2,13,2,1,9,1,6,1,3,1,...]		1.74540566240734686349459630968366106
1.64872 12707 00128 14684	Square root of the number e		$\sqrt {e}$	$\sum_{n = 0}^\infty \frac{1}{2^n n!} = \sum_{n = 0}^\infty \frac{1}{(2n)!!} = \frac{1}{1}+\frac{1}{2}+\frac{1}{8}+\frac{1}{48}+\cdots$	Sum[n=0 to ∞] {1/(2^n n!)}	T	A019774	[1;1,1,1,5,1,1,9,1,1,13,1,1,17,1,1,21,1,1,...] = [1;1,1,1,4p+1], p∈ℕ		1.64872127070012814684865078781416357
1.01734 30619 84449 13971	Zeta(6)	100px	$\zeta(6)$	$\frac{\pi^6}{945} \! = \! \prod_{n=1}^\infty \! \underset{p_n: \text{ prime}}{ \frac{1}{{1-p_n}^{-6}}} = \frac{1}{1 \! -\! 2^{-6}} \! \cdot \! \frac{1}{1 \! - \! 3^{-6}} \! \cdot \! \frac{1}{1 \! - \! 5^{-6}} \cdots$	Prod[n=1 to ∞] {1/(1-ithprime (n)^-6)}	T	A013664	[1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...]		1.01734306198444913971451792979092052
0.10841 01512 23111 36151	Trott constant		$\mathrm{T}_1$	$\textstyle [1, 0, 8, 4, 1, 0, 1, 5, 1, 2, 2, 3, 1, 1, 1, 3, 6,...]$ $\tfrac 1{1+\tfrac 1{0+\tfrac 1{8+\tfrac 1{4+\tfrac 1{1+\tfrac 1{0+1{/\cdots}}}}}}}$			A039662	[0;9,4,2,5,1,2,2,3,1,1,1,3,6,1,5,1,1,2,...]		0.10841015122311136151129081140641509
0.00787 49969 97812 3844	Chaitin constant	40px	$\Omega$	$\sum_{p \in P} 2^{-|p|}$ p: Halted program | p |: Size in bits of program p P: Domain of all programs that stop. See also: Halting problem		T	A100264	[0; 126, 1, 62, 5, 5, 3, 3, 21, 1, 4, 1]	1975	0.0078749969978123844
0.83462 68416 74073 18628	Gauss constant		${G}$	$\frac{1}{\mathrm{agm}(1, \sqrt{2})} = \frac{4 \sqrt{2} \,(\tfrac14 !)^2}{\pi ^{3/2}}= \frac{2}{\pi}\int_0^1\frac{dx}{\sqrt{1 - x^4}}$ AGM = Arithmetic–geometric mean	(4 sqrt(2)((1/4)!)^2) /pi^(3/2)	T	A014549	[0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...]		0.83462684167407318628142973279904680
1.45136 92348 83381 05028	Ramanujan–Soldner constant	100px	${\mu}$	$\mathrm{li}(x) = \int\limits_0^x \frac{dt}{\ln t} = 0 {\color{White}{......}}$ li = Logarithmic integral $\mathrm{li}(x)\;=\;\mathrm{Ei}(\ln{x}) {\color{White}{........}}$ Ei = Exponential integral	FindRoot[li(x) = 0]	I	A070769	[1;2,4,1,1,1,3,1,1,1,2,47,2,4,1,12,1,1,2,2,1,...]	1792 to 1809	1.45136923488338105028396848589202744
0.64341 05462 88338 02618	Cahen's constant		$\xi _{2}$	$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{s_k-1} = \frac{1}{1} - \frac{1}{2} + \frac{1}{6} - \frac{1}{42} + \frac{1}{1806} {\,\pm \cdots}$ Where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ... Defined as: $\, \, S_0= \, 2 , \,\, S_k= \, 1+\prod \limits_{n=0}^{k-1} S_n\text{ for}\;k>0$		T	A080130	[0; 1, 1, 1, 4, 9, 196, 16641, 639988804, ...]	1891	0.64341054628833802618225430775756476
1.41421 35623 73095 04880	Square root of 2, Pythagoras constant.		$\sqrt{2}$	$\! \prod_{n=1}^\infty \! \left( 1 \! + \! \frac{(-1)^{n+1}}{2n-1} \right) \! = \! \left(1 \! + \! \frac{1}{1}\right) \! \left(1 \! - \! \frac{1}{3} \right) \! \left(1 \! + \! \frac{1}{5} \right) \cdots$	prod[n=1 to ∞] {1+(-1)^(n+1) /(2n-1)}	A	A002193	[1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...] = [1;2...]		1.41421356237309504880168872420969808
1.77245 38509 05516 02729	Carlson–Levin constant		${\Gamma}(\tfrac12)$	$\sqrt{\pi} = \left(-\frac{1}{2}\right)! = \int_{-\infty }^{\infty } \frac {1}{e^{x^2}} \, dx = \int_{0 }^{1} \frac {1}{\sqrt{-\ln x}} \, dx$	sqrt (pi)	T	A002161	[1;1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,...]		1.77245385090551602729816748334114518
1.05946 30943 59295 26456	Musical interval between each half tone	100px	$\sqrt[12]{2}$	$\begin{array}{l|ccccccccccccr} \! 2^\frac{x}{12} \! & \!\!\scriptstyle{0} & \!\!\!\!\scriptstyle{1} & \!\!\scriptstyle{2} & \!\!\scriptstyle{3} & \!\!\scriptstyle{4} & \scriptstyle{5} & \!\!\scriptstyle{6} & \!\!\scriptstyle{7} & \!\!\scriptstyle{8} & \!\!\scriptstyle{9} & \!\! \scriptstyle{10} & \!\! \scriptstyle{11} & \!\! \scriptstyle{12} \\ \hline \! \scriptstyle{\textrm{Key}} \! & \!\scriptstyle{\mathrm{C_1}} & \!\!\scriptstyle{\mathrm{C^\#}} & \!\!\scriptstyle{\mathrm{D}} & \!\scriptstyle{\mathrm{D^\#}} & \!\!\scriptstyle{\mathrm{E}} & \scriptstyle{\mathrm{F}} & \!\scriptstyle{\mathrm{F^\#}} & \!\!\scriptstyle{\mathrm{G}} & \!\scriptstyle{\mathrm{G^\#}} & \!\!\scriptstyle{\mathrm{A}} & \!\scriptstyle{\mathrm{A^\#}} & \!\!\scriptstyle{\mathrm{B}} & \!\scriptstyle{\mathrm{C_2}} \end{array}$(A = 440 Hz)	2^(1/12)	A	A010774	[1;16,1,4,2,7,1,1,2,2,7,4,1,2,1,60,1,3,1,2,...]		1.05946309435929526456182529494634170
1.01494 16064 09653 62502	Gieseking constant		${\pi \ln \beta}$	$\frac{3\sqrt{3}}{4} \left(1- \sum_{n=0}^\infty \frac{1}{(3n+2)^2}+ \sum_{n=1}^\infty\frac{1}{(3n+1)^2} \right)=$ $\textstyle \frac{3\sqrt{3}}{4} \left( 1 - \frac{1}{2^2} + \frac{1}{4^2}-\frac{1}{5^2}+\frac{1}{7^2}-\frac{1}{8^2}+\frac{1}{10^2} \pm \cdots \right)$.	sqrt(3)*3/4 *(1 -Sum[n=0 to ∞] {1/((3n+2)^2)} +Sum[n=1 to ∞] {1/((3n+1)^2)})		A143298	[1;66,1,12,1,2,1,4,2,1,3,3,1,4,1,56,2,2,11,...]	1912	1.01494160640965362502120255427452028
2.62205 75542 92119 81046	Lemniscate constant	100px	${\varpi}$	$\pi \, {G} = 4 \sqrt{\tfrac2\pi}\,\Gamma{\left(\tfrac54 \right)^2} = \tfrac14 \sqrt{\tfrac{2}{\pi}}\,\Gamma {\left(\tfrac14 \right)^2} = 4 \sqrt{\tfrac2\pi}\left(\tfrac14 !\right)^2$	4 sqrt(2/pi) ((1/4)!)^2	T	A062539	[2;1,1,1,1,1,4,1,2,5,1,1,1,14,9,2,6,2,9,4,1,...]	1798	2.62205755429211981046483958989111941
1.28242 71291 00622 63687	Glaisher–Kinkelin constant		${A}$	$e^{\frac{1}{12}-\zeta^\prime(-1)} = e^{\frac{1}{8}-\frac{1}{2}\sum\limits_{n=0}^\infty \frac{1}{n+1} \sum\limits_{k=0}^n \left(-1\right)^k \binom{n}{k} \left(k+1\right)^2 \ln(k+1)}$	e^(1/12-zeta´{-1})	T ?	A074962	[1;3,1,1,5,1,1,1,3,12,4,1,271,1,1,2,7,1,35,...]		1.28242712910062263687534256886979172
-4.22745 35333 76265 408	Digamma (1/4)	100px	${\psi} (\tfrac14)$	$-\gamma -\frac{\pi}{2} - 3\ln{2} = -\gamma+\sum_{n=0}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+\tfrac14}\right)$	-EulerGamma -\pi/2 -3 log 2		A020777	-[4;4,2,1,1,10,1,5,9,11,1,22,1,1,14,1,2,1,4,...]		-4.2274535333762654080895301460966835
0.28674 74284 34478 73410	Strongly Carefree constant		$K_{2}$	$\prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left( 1-\frac{3 p_n-2}{{p_n}^{3}}\right)} = \frac {6}{\pi ^2}\prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left( 1-\frac{1}{{p_n(p_n+1)}}\right)}$	N[ prod[k=1 to ∞] {1-(3*prime(k)-2) /(prime(k)^3)}]		A065473	[0;3,2,19,3,12,1,5,1,5,1,5,2,1,1,1,1,1,3,7,...]		0.28674742843447873410789271278983845
3.62560 99082 21908 31193	Gamma(1/4)	100px	$\Gamma(\tfrac14)$	$4 \left(\frac{1}{4}\right)! = \left(-\frac{3}{4}\right)!$	4(1/4)!	T	A068466	[3;1,1,1,2,25,4,9,1,1,8,4,1,6,1,1,19,1,1,4,1,...]	1729	3.62560990822190831193068515586767200
1.66168 79496 33594 12129	Somos' quadratic recurrence constant		${\sigma}$	$\prod_{n=1}^\infty n^{{1/2}^n} = \sqrt {1 \sqrt {2 \sqrt{3 \cdots}}} = 1^{1/2} \; 2^{1/4} \; 3^{1/8} \cdots$	prod[n=1 to ∞] {n ^(1/2)^n}	T ?	A065481	[1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...]		1.66168794963359412129581892274995074
0.95531 66181 245092 78163	Magic angle	100px	${\theta_m}$	$\arctan \left(\sqrt{2}\right) = \arccos \left(\sqrt{\tfrac13}\right) \approx \textstyle {54.7356} ^{ \circ }$	arctan(sqrt(2))	T	A195696	[0;1,21,2,1,1,1,2,1,2,2,4,1,2,9,1,2,1,1,1,3,...]		0.95531661812450927816385710251575775
1.78107 24179 90197 98523	Exp.gamma, Barnes G-function		$e^{\gamma}$	$\prod_{n=1}^\infty \frac{e^{\frac{1}{n}}}{1+\tfrac1n} = \prod_{n=0}^\infty \left(\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac{1}{n+1}} =$ $\textstyle \left ( \frac{2}{1} \right )^{1/2} \left (\frac{2^2}{1 \cdot 3} \right )^{1/3} \left (\frac{2^3 \cdot 4}{1 \cdot 3^3} \right )^{1/4} \left (\frac{2^4 \cdot 4^4}{1 \cdot 3^6 \cdot 5} \right )^{1/5}\cdots$	Prod[n=1 to ∞] {e^(1/n)} /{1 + 1/n}		A073004	[1;1,3,1,1,3,5,4,1,1,2,2,1,7,9,1,16,1,1,1,2,...]		1.78107241799019798523650410310717954
0.74759 79202 53411 43517	Rényi's Parking Constant	100px 100px	${m}$	$\int \limits_{0}^{\infty} exp \left(\! -2 \int \limits_{0}^{x} \frac {1-e^{-y}}{y} dy\right)\! dx = {e^{-2 \gamma}} \int \limits_{0}^{\infty} \frac{e^{-2 \Gamma(0,n)}}{n^2}$	[e^(-2*Gamma)] * Int{n,0,∞}[ e^(- 2 *Gamma(0,n)) /n^2]		A050996	[0;1,2,1,25,3,1,2,1,1,12,1,2,1,1,3,1,2,1,43,...]		0.74759792025341143517873094383017817
1.27323 95447 35162 68615	Ramanujan–Forsyth series		$\frac {4}{\pi}$	$\displaystyle \sum \limits_{n=0}^\infty \textstyle \left(\frac{(2n-3)!!}{(2n)!!}\right)^2 = {1 \! + \! \left(\frac {1}{2} \right)^2 \! + \! \left(\frac {1}{2 \cdot 4} \right)^2 \! + \! \left(\frac {1 \cdot 3}{2 \cdot 4 \cdot 6} \right)^2 + \cdots}$	Sum[n=0 to ∞] {[(2n-3)!! /(2n)!!]^2}	I	A088538	[1;3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...]		1.27323954473516268615107010698011489
1.44466 78610 09766 13365	Steiner number, Iterated exponential Constant		$\sqrt[e]{e}$	$e^{\frac{1}{e}}{\color{White}{...........}}$ = Upper Limit of Tetration	e^(1/e)	T	A073229	[1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]		1.44466786100976613365833910859643022
0.69220 06275 55346 35386	Minimum value of función ƒ(x) = xx		${\left(\frac{1}{e}\right)}^\frac{1}{e}$	${e}^{-\frac{1}{e}} {\color{White}{..........}}$ = Inverse Steiner Number	e^(-1/e)		A072364	[0;1,2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]		0.69220062755534635386542199718278976
0.34053 73295 50999 14282	Pólya Random walk constant		${p(3)}$	$1- \!\!\left({3\over(2\pi)^3}\int\limits_{-\pi}^{\pi} \int\limits_{-\pi}^{\pi} \int\limits_{-\pi}^{\pi} {dx\,dy\,dz\over 3-\!\cos x-\!\cos y-\!\cos z}\right)^{\!-1}$ $= 1- 16\sqrt{\tfrac23}\;\pi^3 \left(\Gamma(\tfrac{1}{24})\Gamma(\tfrac{5}{24})\Gamma(\tfrac{7}{24})\Gamma(\tfrac{11}{24})\right)^{-1}$	1-16*Sqrt[2/3]*Pi^3 /(Gamma[1/24] *Gamma[5/24] *Gamma[7/24] *Gamma[11/24])		A086230	[0;2,1,14,1,3,8,1,5,2,7,1,12,1,5,59,1,1,1,3,...]		0.34053732955099914282627318443290289
0.54325 89653 42976 70695	Bloch–Landau constant		${L}$	$= \frac {\Gamma(\tfrac13)\;\Gamma(\tfrac{5}{6})} {\Gamma(\tfrac{1}{6})} = \frac {(-\tfrac23)!\;(-1+\tfrac56)!} {(-1+\tfrac16)!}$	gamma(1/3) *gamma(5/6) /gamma(1/6)		A081760	[0;1,1,5,3,1,1,2,1,1,6,3,1,8,11,2,1,1,27,4,...]	1929	0.54325896534297670695272829530061323
0.18785 96424 62067 12024	MRB Constant, Marvin Ray Burns	100px	$C_{{}_{MRB}}$	$\sum_{n=1}^{\infty} (-1)^n (n^{1/n}-1) = - \sqrt[1]{1} + \sqrt[2]{2} - \sqrt[3]{3} + \cdots$	Sum[n=1 to ∞] {(-1)^n (n^(1/n)-1)}		A037077	[0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...]	1999	0.18785964246206712024851793405427323
1.46707 80794 33975 47289	Porter Constant		${C}$	$\frac{6\ln 2}{\pi ^2} \left(3 \ln 2 + 4 \,\gamma -\frac{24}{\pi ^2} \,\zeta '(2)-2 \right)-\frac{1}{2}$ $\scriptstyle \gamma \, \text{= Euler–Mascheroni Constant} = 0.5772156649\ldots$ $\scriptstyle \zeta '(2) \,\text{= Derivative of }\zeta(2)= - \sum \limits_{n = 2}^{\infty} \frac{\ln n}{n^2} = -0.9375482543\ldots$	6*ln2/pi^2(3*ln2+ 4 EulerGamma- WeierstrassZeta'(2) *24/pi^2-2)-1/2		A086237	[1;2,7,10,1,2,38,5,4,1,4,12,5,1,5,1,2,3,1,...]	1974	1.46707807943397547289779848470722995
4.66920 16091 02990 67185	Feigenbaum constant δ		${\delta}$	$\lim_{n \to \infty}\frac {x_{n+1}-x_n}{x_{n+2}-x_{n+1}} \qquad \scriptstyle x \in (3.8284;\, 3.8495)$ $\scriptstyle x_{n+1}=\,ax_n(1-x_n)\quad \text{or} \quad x_{n+1}=\,a\sin(x_n)$		T	A006890	[4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...]	1975	4.66920160910299067185320382046620161
2.50290 78750 95892 82228	Feigenbaum constant α		$\alpha$	$\lim_{n \to \infty}\frac {d_n}{d_{n+1}}$		T ?	A006891	[2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...]	1979	2.50290787509589282228390287321821578
0.62432 99885 43550 87099	Golomb–Dickman constant		${\lambda}$	$\int \limits_0^\infty \underset{\text{Para } x>2}{\frac{f(x)}{x^2} \, dx} = \int \limits_0^1 e^{\operatorname{Li}(n)} dn \quad \scriptstyle \text{Li: Logarithmic integral}$	N[Int{n,0,1}[e^Li(n)],34]		A084945	[0;1,1,1,1,1,22,1,2,3,1,1,11,1,1,2,22,2,6,1,...]	1930 & 1964	0.62432998854355087099293638310083724
23.14069 26327 79269 0057	Gelfond constant		${e}^{\pi}$	$(-1)^{-i} = i^{-2i} = \sum_{n=0}^\infty \frac{\pi^{n}}{n!} = \frac{\pi^{1}}{1} + \frac{\pi^{2}}{2!} + \frac{\pi^{3}}{3!} + \cdots$	Sum[n=0 to ∞] {(pi^n)/n!}	T	A039661	[23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...]		23.1406926327792690057290863679485474
7.38905 60989 30650 22723	Conic constant, Schwarzschild constant	100px	$e^2$	$\sum_{n = 0}^\infty \frac{2^n}{n!} = 1+2+\frac{2^2}{2!}+\frac{2^3}{3!}+\frac{2^4}{4!}+\frac{2^5}{5!}+\cdots$	Sum[n=0 to ∞] {2^n/n!}	T	A072334	[7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...] = [7,2,1,1,n,4*n+6,n+2], n = 3, 6, 9, etc.		7.38905609893065022723042746057500781
0.35323 63718 54995 98454	Hafner–Sarnak–McCurley constant (1)		${\sigma}$	$\prod_{k=1}^{\infty}\left\{1-[1-\prod_{j=1}^n \underset{p_k: \text{ prime}}{(1-p_k^{-j})]^2}\right\}$	prod[k=1 to ∞] {1-(1-prod[j=1 to n] {1-ithprime(k)^-j})^2}		A085849	[0;2,1,4,1,10,1,8,1,4,1,2,1,2,1,2,6,1,1,1,3,...]	1993	0.35323637185499598454351655043268201
0.60792 71018 54026 62866	Hafner–Sarnak–McCurley constant (2)		$\frac{1}{\zeta(2)}$	$\frac{6}{\pi^2} = \prod_{n = 0}^\infty \underset{p_n: \text{ prime}}{\! \left(\! 1- \frac{1}{{p_n}^2} \! \right)} \! = \! \textstyle \left(1 \! - \! \frac{1}{2^2}\right) \! \left(1 \! - \! \frac{1}{3^2}\right) \! \left(1 \! - \! \frac{1}{5^2}\right)\cdots$	Prod{n=1 to ∞} (1-1/ithprime(n)^2)	T	A059956	[0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...]		0.60792710185402662866327677925836583
0.12345 67891 01112 13141	Champernowne constant	100px	$C_{10}$	$\sum_{n=1}^\infty \; \sum_{k=10^{n-1}}^{10^n-1}\frac{k}{10^{kn-9\sum_{j=0}^{n-1}10^j(n-j-1)}}$		T	A033307	[0;8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15,...]	1933	0.12345678910111213141516171819202123
0.76422 36535 89220 66299	Landau-Ramanujan constant		$K$	$\frac1{\sqrt2}\prod_{p\equiv3\!\!\!\!\!\mod \! 4}\!\! \underset{\!\!\!\!\!\!\!\! p: \text{ prime}}{\left(1-\frac1{p^2}\right)^{-\frac{1}{2}}}\!\!=\frac\pi4\prod_{p\equiv1\!\!\!\!\!\mod \!4}\!\! \underset{\!\!\!\! p: \text{ prime}}{\left(1-\frac1{p^2}\right)^\frac{1}{2}}$		T ?	A064533	[0;1,3,4,6,1,15,1,2,2,3,1,23,3,1,1,3,1,1,6,4,...]		0.76422365358922066299069873125009232
2.71828 18284 59045 23536	Number e, Euler's number		${e}$	$\! \lim_{n \to \infty} \! \left( \! 1 \! + \! \frac {1}{n}\right)^n \! = \! \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \textstyle \cdots$	Sum[n=0 to ∞] {1/n!} (* lim_(n->∞) (1+1/n)^n *)	T	A001113	[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...] = [2;1,2p,1], p∈ℕ		2.71828182845904523536028747135266250
0.36787 94411 71442 32159	Inverse of Number e		$\frac{1}{e}$	$\sum_{n = 0}^\infty \frac{(-1)^n}{n!} = \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} +\cdots$	Sum[n=2 to ∞] {(-1)^n/n!}	T	A068985	[0;2,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] = [0;2,1,1,2p,1], p∈ℕ	1618	0.36787944117144232159552377016146086
0.69034 71261 14964 31946	Upper iterated exponential	100px	${H}_{2n+1}$	$\lim_{n \to \infty} {H}_{2n+1} = \textstyle \left(\frac{1}{2}\right) ^{\left(\frac{1}{3}\right) ^{\left(\frac{1}{4}\right) ^{\cdot^{\cdot^{\left(\frac{1}{2n+1}\right)}}}}} = {2}^{-3^{-4^{\cdot^{\cdot^{{-2n-1}}}}}}$	2^-3^-4^-5^-6^ -7^-8^-9^-10^ -11^-12^-13 …		A242760	[0;1,2,4,2,1,3,1,2,2,1,4,1,2,4,3,1,1,10,1,3,2,...]		0.69034712611496431946732843846418942
0.65836 55992 ...	Lower límit iterated exponential		${H}_{2n}$	$\lim_{n \to \infty} {H}_{2n} = \textstyle \left(\frac{1}{2}\right) ^{\left(\frac{1}{3}\right) ^{\left(\frac{1}{4}\right) ^{\cdot^{\cdot^{\left(\frac{1}{2n}\right)}}}}} = {2}^{-3^{-4^{\cdot^{\cdot^{{-2n}}}}}}$	2^-3^-4^-5^-6^ -7^-8^-9^-10^ -11^-12 …			[0;1,1,1,12,1,2,1,1,4,3,1,1,2,1,2,1,51,2,2,1,...]		0.6583655992...
3.14159 26535 89793 23846	π number, Archimedes number		$\pi$	$\lim_{n\to \infty }\, 2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_n$	Sum[n=0 to ∞] {(-1)^n 4/(2n+1)}	T	A000796	[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...]		3.14159265358979323846264338327950288
1.92878 00...	Wright constant		${\omega}$	$\left \lfloor 2^{2^{2^{\cdot^{\cdot^{2^{\omega}}}}}} \!\right \rfloor \scriptstyle \text{= primes:} \displaystyle\left\lfloor 2^\omega\right\rfloor \scriptstyle \text{=3,} \displaystyle\left\lfloor 2^{2^\omega} \right\rfloor \scriptstyle \text{=13,} \displaystyle \left\lfloor 2^{2^{2^\omega}} \right\rfloor \scriptstyle =16381, \ldots$			A086238	[1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3]		1.9287800...
0.46364 76090 00806 11621	Machin–Gregory series	100px	$\arctan \frac {1}{2}$	$\underset{\text{For } x = 1/2 \qquad \qquad} {\sum_{n=0}^\infty \frac{(\!-1\!)^n \, x^{2n+1}}{2n+1} = \frac {1}{2} {-} \frac{1}{3 \! \cdot \! 2^3} {+} \frac{1}{5 \! \cdot \! 2^5} {-} \frac{1}{7 \! \cdot \! 2^7} {+} \cdots}$	Sum[n=0 to ∞] {(-1)^n (1/2)^(2n+1) /(2n+1)}	I	A073000	[0;2,6,2,1,1,1,6,1,2,1,1,2,10,1,2,1,2,1,1,1,...]		0.46364760900080611621425623146121440
0.69777 46579 64007 98200	Continued fraction constant, Bessel function		${C}_{CF}$	$\frac{I_1(2)}{I_0(2)} = \frac{ \sum \limits_{n = 0}^\infty \frac{n}{n!n!}} {{ \sum \limits_{n = 0}^{\infty} \frac{1}{n!n!}}} = \textstyle \tfrac 1{1+\tfrac 1{2+\tfrac 1{3+\tfrac 1{4+\tfrac 1{5+\tfrac 1{6+1{/\cdots}}}}}}}$	(Sum [n=0 to ∞] {n/(n!n!)}) / (Sum [n=0 to ∞] {1/(n!n!)})	I	A052119	[0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...] = [0;p+1], p∈ℕ		0.69777465796400798200679059255175260
1.90216 05831 04	Brun 2 constant = Σ inverse of Twin primes	100px	${B}_{\,2}$	$\textstyle \underset{ p,\, p+2: \text{ prime}}{\sum(\frac1{p}+\frac1{p+2})} = (\frac1{3} \! + \! \frac1{5}) + (\tfrac1{5} \! + \! \tfrac1{7}) + (\tfrac1{11} \! + \! \tfrac1{13}) + \cdots$			A065421	[1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2]		1.902160583104
0.87058 83799 75	Brun 4 constant = Σ inv.prime quadruplets		${B}_{\,4}$	$\textstyle {\sum(\frac1{p}+\frac1{p+2}+\frac1{p+6}+\frac1{p+8})} \scriptstyle \quad {p,\; p+2,\; p+6,\; p+8: \text{ prime}}$ $\textstyle{\left(\tfrac1{5} + \tfrac1{7} + \tfrac1{11} + \tfrac1{13}\right)}+ \left(\tfrac1{11} + \tfrac1{13} + \tfrac1{17} + \tfrac1{19}\right)+ \dots$			A213007	[0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1]		0.870588379975
0.63661 97723 67581 34307	Buffon constant	100px	$\frac{2}{\pi}$	$\frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots$ Viète product	2/Pi	T	A060294	[0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...]	1540 to 1603	0.63661977236758134307553505349005745
0.59634 73623 23194 07434	Euler–Gompertz constant		${G}$	$\! \int \limits_0^\infty \!\! \frac{e^{-n}}{1{+}n} \, dn = \!\! \int \limits_0^1 \!\! \frac{1}{1{-}\ln n} \, dn = \textstyle {\tfrac 1 {1+\tfrac 1{1+\tfrac 1{1+\tfrac 2{1+\tfrac 2{1+\tfrac 3{1+3{/\cdots}} }}}}}}$	integral[0 to ∞] {(e^-n)/(1+n)}	I	A073003	[0;1,1,2,10,1,1,4,2,2,13,2,4,1,32,4,8,1,1,1,...]		0.59634736232319407434107849936927937
i ···	Imaginary number	100px	${i}$	$\sqrt{-1} = \frac{\ln(-1)}{\pi} \qquad\qquad \mathrm{e}^{i\,\pi} = -1$	sqrt(-1)	C I			1501 to 1576	i
2.74723 82749 32304 33305	Ramanujan nested radical		$R_{5}$	$\scriptstyle \sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+ \sqrt{5+\sqrt{5+\sqrt{5-\cdots}}}}}}}\;= \textstyle\frac{2+\sqrt{5}+\sqrt{15-6\sqrt{5}}}{2}$	(2+sqrt(5) +sqrt(15 -6 sqrt(5)))/2	A		[2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...]		2.74723827493230433305746518613420282
0.56714 32904 09783 87299	Omega constant, Lambert W function		${\Omega}$	$\sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!} =\,\left(\frac{1}{e}\right) ^{\left(\frac{1}{e}\right) ^{\cdot^{\cdot^{\left(\frac{1}{e}\right)}}}} = e^{-\Omega} = e^{-e^{-e^{\cdot^{\cdot^{{-e}}}}}}$	Sum[n=1 to ∞] {(-n)^(n-1)/n!}	T	A030178	[0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,2,1,...]		0.56714329040978387299996866221035555
0.96894 61462 59369 38048	Beta(3)		${\beta} (3)$	$\frac{\pi^3}{32} = \sum_{n=1}^\infty\frac{-1^{n+1}}{(-1+2n)^3} = \frac{1}{1^3} {-} \frac{1}{3^3} {+} \frac{1}{5^3} {-} \frac{1}{7^3} {+} \cdots$	Sum[n=1 to ∞] {(-1)^(n+1) /(-1+2n)^3}	T	A153071	[0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...]		0.96894614625936938048363484584691860
2.23606 79774 99789 69640	Square root of 5, Gauss sum		$\sqrt{5}$	$\scriptstyle (n = 5) \displaystyle \sum_{k=0}^{n-1} e^{\frac{2 k^2 \pi i}{n}} = 1 + e^\frac{2 \pi i} {5} + e^\frac{8 \pi i} {5} + e^\frac{18 \pi i} {5} + e^\frac{32 \pi i} {5}$	Sum[k=0 to 4] {e^(2k^2 pi i/5)}	A	A002163	[2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...] = [2;4,...]		2.23606797749978969640917366873127624
3.35988 56662 43177 55317	Prévost constant Reciprocal Fibonacci constant		$\Psi$	$\sum_{n=1}^{\infty} \frac{1}{F_n} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \cdots$ Fn: Fibonacci series	Sum[n=1 to ∞] {1/Fibonacci[n]}	I	A079586	[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...]	?	3.35988566624317755317201130291892717
2.68545 20010 65306 44530	Khinchin's constant	100px	$K_{\,0}$	$\prod_{n=1}^\infty \left[{1+{1\over n(n+2)}}\right]^{\ln n/\ln 2}$	Prod[n=1 to ∞] {(1+1/(n(n+2))) ^(ln(n)/ln(2))}	T	A002210	[2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...]	1934	2.68545200106530644530971483548179569

See also

• Invariant (mathematics)
• List of numbers
• Mathematical constant

External links

• Inverse Symbolic Calculator, Plouffe's Inverter
• Constants - from Wolfram MathWorld
• On-Line Encyclopedia of Integer Sequences (OEIS)
• Steven Finch's page of mathematical constants
• Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms