# قائمة الثوابت الرياضية

الثابت الرياضي هو رقم، له دلالة خاصة في العمليات الحسابية. على سبيل المثال، الثابت الرياضي باي (π) يعني نسبة طول محيط الدائرة إلى قطرها. هذه القيمة ثابته لا تتغير لأي دائرة.

## بيانات الجدول

* القيمة العددية للثابت: من المراجع العالمية كموقع ماثوورلد أو من موسوعة المتتاليات الصحيحة على الإنترنت والتي تعرف اختصارا باسم أويس ويكي (OEIS).
* لاتخ: الصيغة في تنسيق تخ.
* الصيغة الرياضية: المستخدمة في برنامج ولفرام ألفا.
* أويس: موسوعة على الإنترنت عن الأعداد الصحيحة.
* الكسر المستمر: وهو الكسر الذي يأخذ الصيغة التالية
    $x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots \,}}}}}}}}$
* العام: عام اكتشاف الثابت.
* تنسيق الويب: القيمة بشكل مناسب على صفحات الإنترنت.
* النوع: نوع العدد.
** ك: عدد كسري
** غ.ك: عدد غير نسبي (عدد غير كسري)
** ج : عدد جبري
** م : عدد متسام
** خ : عدد مركب

## جدول الدوال والثوابت الرياضية

هذا الجدول يهتم بأهم الدوال والثوابت الرياضية على مر العصور:
| القيمة العددية | الاسم | الرسومات | الرمز | لاتخ | الصيغة | النوع | أويس ويكي | الكسر المستمر | العام | تنسيق الويب |
| 0.74048048969306104116 | ثابت هيرميت تعبئة الكرات بنظام ثلاثي الأبعاد حدسية كيبلر | | ${\mu _{_{K}}}$ | ${\frac {\pi }{3{\sqrt {2}}}}{\color {white}......\color {black}}$ أثبت توماس هيلز في عام 2014 أن حدثية كيبلر صحيحة. | 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 |
| 22.45915771836104547342 | 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 |
| 2.80777024202851936522 | ثابت فرانسين روبنسون | | ${F}$ | $\int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx.=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx$ | 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 |
| 1.305686729 ≈ بواسطة توماس ودهار 1.305688 ≈ بواسطة ماكمولين | الهندسة الكسيرية لأبلونيوس البرغاوي · | | $\varepsilon$ | | | | 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 ≈ |
| 0.43828293672703211162 0.360592471871385485 i | الأس الانهائي للوحدة التخليلةi | | $^{\infty }{i}$ | $\lim _{n\to \infty }^{n}i=\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}$ | i^i^i^i^i^i^... | خ | 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 |
| 0.9288358271 | مجموع مقلوب الأعداد الأولية التوأم | | $B_{1}$ | ${\frac {1}{4}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{18}}+{\frac {1}{30}}+{\frac {1}{42}}+{\frac {1}{60}}+{\frac {1}{72}}+\cdots$ | 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 |
| 0.63092975357145743709 | مجموعة كانتور | | $d_{f}(k)$ | $\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}={\frac {\log 2}{\log 3}}$ | log(2)/log(3) N[3^x=2] | م | A102525 | [0;1,1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...] | | 0.63092975357145743709952711434276085 |
| 0.31830988618379067153 | مقلوب باي (π), سرينفاسا أينجار رامانجن | | ${\frac {1}{\pi }}$ | ${\frac {2{\sqrt {2}}}{9801}}\sum _{n=0}^{\infty }{\frac {(4n)!\,(1103+26390\;n)}{(n!)^{4}\,396^{4n}}}$ | 2 sqrt(2)/9801 * Sum[n=0 to ∞] {((4n)!/n!^4) *(1103+ 26390n) / 396^(4n)} | م | A049541 | [0;3,7,15,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...] | | 0.31830988618379067153776752674502872 |
| 0.28878809508660242127 | فلاجوليت وريتشموند | | ${Q}$ | $\prod _{n=1}^{\infty }\left(1-{\frac {1}{2^{n}}}\right)=\left(1{-}{\frac {1}{2^{1}}}\right)\left(1{-}{\frac {1}{2^{2}}}\right)\left(1{-}{\frac {1}{2^{3}}}\right)...$ | 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 |
| 1.53960071783900203869 | ثابت إليوت هرشل ليب للجليد (يستخدم في تحديد عدد المسارات الاويلرية) | | ${W}_{2D}$ | $\lim _{n\to \infty }(f(n))^{n^{-2}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}$ | (4/3)^(3/2) | ج | A118273 | [1;1,1,5,1,4,2,1,6,1,6,1,2,4,1,5,1,1,2,...] | 1967 | 1.53960071783900203869106341467188655 |
| 0.20787957635076190854 | $i^{i}$ | | $i^{i}$ | $e^{-{\frac {\pi }{2}}}$ | e^(-π/2) | م | A049006 | [0;4,1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,...] | 1746 | 0.20787957635076190854695561983497877 |
| 4.53236014182719380962 | ثابت فان دير باو | | ${\alpha }$ | ${\frac {\pi }{\ln(2)}}={\frac {\sum \limits _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}}{\sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}}}={\frac {{\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-\cdots }{{\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }}$ | 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 |
| 0.76159415595576488811 | دالة زائدية للعدد 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) | م | A073744 | [0;1,3,5,7,9,11,13,15,17,19,21,23,25,27,...] = [0;2p+1], p∈ℕ | | 0.76159415595576488811945828260479359 |
| 0.59017029950804811302 | ثابت تشيبيشيف · | | ${\lambda _{Ch}}$ | ${\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}={\frac {4({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}$ | (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 |
| 0.07077603931152880353 0.6840003894379- | MKB ثابت · · | | $M_{I}$ | $\lim _{n\rightarrow \infty }\int _{1}^{2n}(-1)^{x}~{\sqrt[{x}]{x}}~dx=\int _{1}^{2n}e^{i\pi x}~x^{1/x}~dx$ | lim_(2n->∞) int[1 to 2n] {exp(i*Pi*x)*x^(1/x) dx} | خ | 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 |
| 1.259921049894873164767 | الجذر التكعيبي للرقم 2 | | ${\sqrt[{3}]{2}}$ | ${\sqrt[{3}]{2}}$ | 2^(1/3) | ج | A002580 | [1;3,1,5,1,1,4,1,1,8,1,14,1,10,2,1,4,12,2,3,...] | | 1.25992104989487316476721060727822835 |
| 1.09317045919549089396 | ثابت سمراندش 1ª | | ${S_{1}}$ | $\sum _{n=2}^{\infty }{\frac {1}{\mu (n)!}}{\color {white}....\color {black}}$ حيث μ(n) هو دالة كيمبنر | | | A048799 | [1;10,1,2,1,2,1,13,3,1,6,1,2,11,4,6,2,15,1,1,...] | | 1.09317045919549089396820137014520832 |
| 0.62481053384382658687 + 1.30024 25902 20120 419 i | الكسر المستمر المعمم للوحدة التخليلية i | | ${{F}_{CG}}_{(i)}$ | $\textstyle i{+}{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+i{/...}}}}}}}}}}}}}={\sqrt {\frac {{\sqrt {17}}-1}{8}}}+i\left({\tfrac {1}{2}}{+}{\sqrt {\frac {2}{{\sqrt {17}}-1}}}\right)$ | 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/( ...))))))))))))))))))))) | ج | 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 |
| 3.05940740534257614453 | ثابت المضروب المزدوج | | ${C_{_{n!!}}}$ | $\sum _{n=0}^{\infty }{\frac {1}{n!!}}={\sqrt {e}}\left[{\frac {1}{\sqrt {2}}}+\gamma ({\tfrac {1}{2}},{\tfrac {1}{2}})\right]$ | 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 |
| 5.97798681217834912266 | ثابت ماديلونغ | | ${H}_{2}(2)$ | $\pi \ln(3){\sqrt {3}}$ | 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 |
| 0.91893853320467274178 | صيغة راب | | ${\zeta '(0)}$ | $\int \limits _{a}^{a+1}\log \Gamma (t)\,\mathrm {d} t={\tfrac {1}{2}}\log 2\pi +a\log a-a,\quad a\geq 0$ | 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 |
| 2.20741609916247796230 | مسألة الأريكة المتحركة | | ${S_{_{H}}}$ | ${\frac {\pi }{2}}+{\frac {2}{\pi }}\,$ | pi/2 + 2/pi | م | 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 |
| 1.17628081825991750654 | عدد سالم، تخيل ليمير | | ${\sigma _{_{10}}}$ | $x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1$ | x^10+x^9-x^7-x^6 -x^5-x^4-x^3+x+1 | ج | A073011 | [1;5,1,2,17,1,7,2,1,1,2,4,7,2,2,1,1,15,1,1, ... | 1983? | 1.17628081825991750654407033847403505 |
| 0.37395581361920228805 | ثابت إميل أرتين | | ${C}_{Artin}$ | $\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}-1)}}\right)\quad p_{n}\scriptstyle {\text{ = prime}}$ | 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 |
| 0.42215773311582662702 | حجم رباعي الأسطح | | ${V_{_{R}}}$ | ${\frac {s^{3}}{12}}(3{\sqrt {2}}-49\,\pi +162\,\arctan {\sqrt {2}})$ | (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 |
| 2.82641999706759157554 | ثابت موراتا | | ${C_{m}}$ | $\prod _{n=1}^{\infty }{\underset {p_{n}:\,{prime}}{{\Big1+{\frac {1}{(p_{n}-1)^{2}}}{\Big )}}}$ | 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 |
| 1.09864196439415648573 | ثابت باريس | | $C_{Pa}$ | $\prod _{n=2}^{\infty }{\frac {2\varphi }{\varphi +\varphi _{n}}}\;,\;\varphi {=}{Fi}$ 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 |
| 2.39996322972865332223 بالراديان | الزاوية الذهبية | | ${b}$ | $(4-2\,\Phi )\,\pi =(3-{\sqrt {5}})\,\pi$ = 137.5077640500378546 ...° | (4-2*Phi)*Pi | م | A131988 | [2;2,1,1,1087,4,4,120,2,1,1,2,1,1,7,7,2,11,...] | 1907 | 2.39996322972865332223155550663361385 |
| 1.64218843522212113687 | ثابت ليبيسج | | ${L2}$ | ${\frac {1}{5}}+{\frac {\sqrt {25-2{\sqrt {5}}}}{\pi }}={\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\left|\sin({\frac {5t}{2}})\right|}{\sin({\frac {t}{2}})}}\,dt$ | 1/5 + sqrt(25 - 2*sqrt(5))/Pi | م | 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 |
| 1.26408473530530111307 | ثابت فارديt | | ${V_{c}}$ | ${\frac {\sqrt {3}}{\sqrt {2}}}\prod _{n\geq 1}\left(1+{1 \over (2e_{n}-1)^{2}}\right)^{\!1/2^{n+1}}$ | | | 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 | مساحة مجموعة ماندلبرو | | $\gamma$ | ${\sqrt {6\pi -1}}-e=1.506591651\cdots$ | | | 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 |
| 1.6111149258083 | ثابت المضروب الأسي | | ${S_{Ef}}$ | $\sum _{n=1}^{\infty }{\frac {1}{n^{(n{-}1)^{\cdot ^{\cdot ^{\cdot ^{2^{1}}}}}}}}=1{+}{\frac {1}{2^{1}}}{+}{\frac {1}{3^{2^{1}}}}+{\frac {1}{4^{3^{2^{1}}}}}+{\frac {1}{5^{4^{3^{2^{1}}}}}}{+}\cdots$ | | م | A080219 | [1; 1, 1, 1, 1, 2, 1, 808, 2, 1, 2, 1, 14,...] | | 1.61111492580837673611111111111111111 |
| 1.11786415118994497314 | ثابت جوه شموتز | | $C_{GS}$ | $\int _{0}^{\infty }{\frac {\log(s+1)}{e^{s}-1}}\ ds=\!-\!\sum _{n=1}^{\infty }{\frac {e^{n}}{n}}Ei(-n)$ | 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 |
| 0.3181315052047641 ±1.337235701430689 | النقط الثابتة على اللوغاريتم الأكبر · | | ${-W(-1)}$ | $\lim _{n\rightarrow \infty }$$f(x)=\underbrace {\log(\log(\log(\log(\cdots \log(\log(x))))))\,\!} \atop {\log _{s}{\text{ n times}}}$ تختلف القيمة الابتدائية لx لتصبح 0, 1, e, e$^e$, e$^{ee}${\textstyle 0, 1, e, e^{e}, e^{e^{e}}}, etc. | -W(-1) | خ | A059526  A059527 | [-i;1 +2i,1+i,6-i,1+2i,-7+3i,2i,2,1-2i,-1+i,-, ...] | | 0.31813150520476413531265425158766451 -1.33723570143068940890116214319371 i |
| 0.28016949902386913303 | ثابت بيرنشتين | | ${\beta }$ | $\approx {\frac {1}{2{\sqrt {\pi }}}}$ | 1/(2 sqrt(pi)) | م | A073001 | [0;3,1,1,3,9,6,3,1,3,14,34,2,1,1,60,2,2,1,1,...] | 1913 | 0.28016949902386913303643649123067200 |
| 0.66016181584686957392 | ثابت العددان الأوليان التوأمان | | ${C}_{2}$ | $\prod _{p=3}^{\infty }{\frac {p(p-2)}{(p-1)^{2}}}$ | 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 |
| 1.22674201072035324441 | ثابت معامل فيبوناتشي | | $F$ | $\prod _{n=1}^{\infty }\left(1-\left(-{\frac {1}{{\varphi }^{2}}}\right)^{n}\right)=\prod _{n=1}^{\infty }\left(1-\left({\frac {{\sqrt {5}}-3}{2}}\right)^{n}\right)$ | 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 |
| 0.11494204485329620070 | ثابت كيبلر-بووكمب | | ${\rho }$ | $\prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...$ | 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 |
| 1.78723165018296593301 | ثابت كومورنيك-لوريتي | | ${q}$ | $1=\!\sum _{n=1}^{\infty }{\frac {t_{k}}{q^{k}}}\qquad \scriptstyle {\text{Raiz real de}}\displaystyle \prod _{n=0}^{\infty }\!\left(\!1{-}{\frac {1}{q^{2^{n}}}}\!\right)\!{+}{\frac {q{-}2}{q{-}1}}=0$ | FindRoot[(prod[n=0 to ∞] {1-1/(x^2^n)}+(x-2) /(x-1))= 0, {x, 1.7}, WorkingPrecision->30] | م | A055060 | [1;1,3,1,2,3,188,1,12,1,1,22,33,1,10,1,1,7,...] | 1998 | 1.78723165018296593301327489033700839 |
| 3.30277563773199464655 | القيمة البرونزية | | ${\sigma }_{\,Rr}$ | ${\frac {3+{\sqrt {13}}}{2}}=1+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+\cdots }}}}}}}}$ | (3+sqrt 13)/2 | ج | A098316 | [3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...] = [3;3,...] | | 3.30277563773199464655961063373524797 |
| 0.82699334313268807426 | تغطية القرص | | ${C_{5}}$ | ${\frac {1}{\sum \limits _{n=0}^{\infty }{\frac {1}{\binom {3n+2}{2}}}}}={\frac {3{\sqrt {3}}}{2\pi }}$ | 3 Sqrt[3]/(2 Pi) | م | 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 |
| 2.66514414269022518865 | ثابتة غيلفوند–شنايدر | | $G_{\,GS}$ | $2^{\sqrt {2}}$ | 2^sqrt{2} | م | A007507 | [2;1,1,1,72,3,4,1,3,2,1,1,1,14,1,2,1,1,3,1,...] | 1934 | 2.66514414269022518865029724987313985 |
| 3.27582291872181115978 | ثابت ليفي | | $\gamma$ | $e^{\pi ^{2}/(12\ln 2)}$ | 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 |
| 0.52382257138986440645 | دالة تشي | | ${\operatorname {Chi} }$ | $\gamma +\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt$ $\scriptstyle \gamma \,{\text{= 0.5772156649...}}$ | 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 |
| 1.1319882487943 | ثابت فيسونث | | ${C}_{Vi}$ | $\lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}$  حيثa$_n$ = عدد فيبوناتشي | lim_(n->∞) |a_n|^(1/n) | م | A078416 | [1;7,1,1,2,1,3,2,1,2,1,8,1,5,1,1,1,9,1,...] | 1997 | 1.1319882487943 |
| 1.23370055013616982735 | ثابت فاراد | | ${\tfrac {3}{4}}\zeta (2)$ | ${\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$ | sum[n=1 to ∞] {1/((2n-1)^2)} | م | 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 |
| 2.50662827463100050241 | الجذر التربيعي ل 2 باي | | ${\sqrt {2\pi }}$ | ${\sqrt {2\pi }}=\lim _{n\to \infty }{\frac {n!\;e^{n}}{n^{n}{\sqrt {n}}}}{\color {white}....\color {black}}$ تقريب ستيرلينغ | sqrt (2 pi) | م | 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 |
| 4.13273135412249293846 | الجذر التربيعي لتاو* مشتقة الدالة الأسية للأساس 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 |
| 0.97027011439203392574 | ثابت لوتش | | ${{\text{£}}_{_{Lo}}}$ | ${\frac {6\ln 2\ln 10}{\pi ^{2}}}$ | 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 |
| 0.98770039073605346013 | المساحة المحيطة لمثلث رولو | | ${\mathcal {T}}_{R}$ | $a^{2}\cdot \left(2{\sqrt {3}}+{\frac {\pi }{6}}-3\right)$  حيث a= طول ضلع المربع | 2 sqrt(3)+pi/6-3 | م | 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 |
| 0.70444220099916559273 | ثابت الإهمال $_2$ | | ${\mathcal {C}}_{2}$ | ${\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}+1)}}\right)}}$ | 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 |
| 1.84775906502257351225 | معامل الربط | | ${\mu }$ | ${\sqrt {2+{\sqrt {2}}}}\;=\lim _{n\rightarrow \infty }c_{n}^{1/n}$ دالة متعددة الحدود: $\;x^{4}-4x^{2}+2=0$ | sqrt(2+sqrt(2)) | ج | A179260 | [1;1,5,1,1,3,6,1,3,3,10,10,1,1,1,5,2,3,1,1,3,...] | | 1.84775906502257351225636637879357657 |
| 0.30366300289873265859 | ثابت جاووس-كوزمين-يرسينغ | | ${\lambda }_{2}$ | $\lim _{n\to \infty }{\frac {F_{n}(x)-\ln(1-x)}{(-\lambda )^{n}}}=\Psi (x),$ حيث $\Psi (x)$ دالة تحليلية و $\Psi (0)\!=\!\Psi (1)\!=\!0$. | | | 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 |
| 1.57079632679489661923 | ثابت فارد K1 جداء واليس | | ${\frac {\pi }{2}}$ | $\prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right)={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots$ | Prod[n=1 to ∞] {(4n^2)/(4n^2-1)} | م | 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 |
| 1.606695152415291763 | ثابت إيردوس بروين | | ${E}_{\,B}$ | $\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!...$ | sum[n=1 to ∞] {1/(2^n-1)} | غ.ك | A065442 | [1;1,1,1,1,5,2,1,2,29,4,1,2,2,2,2,6,1,7,1,...] | 1949 | 1.60669515241529176378330152319092458 |
| 1.61803398874989484820 | فاي، النسبة الذهبية | | ${\varphi }$ | ${\frac {1+{\sqrt {5}}}{2}}={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}$ | (1+5^(1/2))/2 | ج | 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 |
| 1.64493406684822643647 | دالة ريمان زيتا (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} | م | 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 |
| 1.73205080756887729352 | الجذر التربيعي ل 3 | | ${\sqrt {3}}$ | ${\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,\cdots }}}}}}}}}}$ | (3(3(3(3(3(3(3) ^1/3)^1/3)^1/3) ^1/3)^1/3)^1/3) ^1/3 ... | ج | 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 |
| 1.75793275661800453270 | عدد كاسنر | | ${R}$ | ${\sqrt {1+{\sqrt {2+{\sqrt {3+{\sqrt {4+\cdots }}}}}}}}$ | 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 |
| 2.29558714939263807403 | ثابت القطع المكافئ العالمي | | ${P}_{\,2}$ | $\ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arcsinh} (1)+{\sqrt {2}}$ | ln(1+sqrt 2)+sqrt 2 | م | 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 |
| 1.78657645936592246345 | ثابت سيلفرمان | | ${{\mathcal {S}}_{_{m}}}$ | $\sum _{n=1}^{\infty }{\frac {1}{\phi (n)\sigma _{1}(n)}}={\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1+\sum _{k=1}^{\infty }{\frac {1}{p_{n}^{2k}-p_{n}^{k-1}}}\right)}}$ ø = مؤشر أويلر، σ$_1$ = دالة القواسم. | 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 |
| 2.59807621135331594029 | مساحة شكل سداسي منتظم مع جانب يساوي 1 | | ${\mathcal {A}}_{6}$ | ${\frac {3{\sqrt {3}}}{2}}$ | 3 sqrt(3)/2 | ج | 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 |
| 0.66131704946962233528 | ثابت فيلر تورنر | | ${{\mathcal {C}}_{_{FT}}}$ | ${\underset {p_{n}:\,{prime}}{{\frac {1}{2}}\prod _{n=1}^{\infty }\left(1-{\frac {2}{p_{n}^{2}}}\right){+}{\frac {1}{2}}}}={\frac {3}{\pi ^{2}}}\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}^{2}-1}}\right){+}{\frac {1}{2}}$ | [prod[n=1 to ∞] {1-2/prime(n)^2}] /2 + 1/2 | م | A065493 | [0;1,1,1,20,9,1,2,5,1,2,3,2,3,38,8,1,16,2,2,...] | 1932 | 0.66131704946962233528976584627411853 |
| 1.46099848620631835815 | ثابت باكستر | Mapamundi 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 = دالة غاما | 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 |
| 1.92756197548292530426 | ثابت تترنك | | ${\mathcal {T}}$ | الجذور الموجبة للمعادلة التالية: $\;\;x^{4}-x^{3}-x^{2}-x-1=0$ | Root[x+x^-4-2=0] | ج | A086088 | [1;1,12,1,4,7,1,21,1,2,1,4,6,1,10,1,2,2,1,7,1,...] | | 1.92756197548292530426190586173662216 |
| 1.00743475688427937609 | مكعب روبرت الرايني | | ${f_{(3,4)}}$ | الجذور الموجبة للمعادلة التالية: $\;\;4x^{4}{-}28x^{3}{-}7x^{2}{+}16x{+}16=0$ | Root[4*x^8-28*x^6 -7*x^4+16*x^2+16 =0] | ج | A243309 | [1;134,1,1,73,3,1,5,2,1,6,3,11,4,1,5,5,1,1,48,...] | | 1.00743475688427937609825359523109914 |
| 1.70521114010536776428 | ثابت نيفن | | ${C}$ | $1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)$ | 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 |
| 0.6045997880780726168 | العلاقة بين مساحة مثلث متساوي الأضلاع والدائر بداخلة | | ${\frac {\pi }{3{\sqrt {3}}}}$ | $\sum _{n=1}^{\infty }{\frac {1}{n{2n \choose n}}}=1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{8}}+\cdots$ متسلسلة دركليه | Sum[1/(n Binomial[2 n, n]) , {n, 1, ∞}] | م | A073010 | [0;1,1,1,1,8,10,2,2,3,3,1,9,2,5,4,1,27,27,6,6,...] | | 0.60459978807807261686469275254738524 |
| 1.15470053837925152901 | ثابت هيرمت | | $\gamma _{_{2}}$ | ${\frac {2}{\sqrt {3}}}={\frac {1}{\cos \,({\frac {\pi }{6}})}}$ | 2/sqrt(3) | ج | 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 |
| 0.41245403364010759778 | ثابت موروس | | $\tau$ | $\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}$  حيث $\tau (x)=\sum _{n=0}^{\infty }(-1)^{t_{n}}\,x^{n}=\prod _{n=0}^{\infty }(1-x^{2^{n}})$ | | م | 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 |
| 0.58057755820489240229 | ثابت بيل | | ${{\mathcal {P}}_{_{Pell}}}$ | $1-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2n+1}}}\right)$ | N[1-prod[n=0 to ∞] {1-1/(2^(2n+1)}] | م | A141848 | [0;1,1,2,1,1,1,1,14,1,3,1,1,6,9,18,7,1,27,1,1,...] | | 0.58057755820489240229004389229702574 |
| 0.66274341934918158097 | نهاية لابلاس | | ${\lambda }$ | ${\frac {x\;e^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1$ | (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 |
| 0.17150049314153606586 | ثابت هال مونتغمري | | ${{\delta }_{_{0}}}$ | $1+{\frac {\pi ^{2}}{6}}+2\;\mathrm {Li} _{2}\left(-{\sqrt {e}}\;\right)\quad \mathrm {Li} _{2}\,\scriptstyle {\text{= Dilogarithm integral}}$ | 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 |
| 1.55138752454832039226 | مثلث كالبي | | ${C_{_{CR}}}$ | ${1 \over 3}+{(-23+3i{\sqrt {237}})^{\tfrac {1}{3}} \over 3\cdot 2^{\tfrac {2}{3}}}+{11 \over 3(2(-23+3i{\sqrt {237}}))^{\tfrac {1}{3}}}$ | FindRoot[ 2x^3-2x^2-3x+2 ==0, {x, 1.5}, WorkingPrecision->40] | ج | 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 |
| 1.22541670246517764512 | غاما(3/4) | | $\Gamma ({\tfrac {3}{4}})$ | $\left(-1+{\frac {3}{4}}\right)!=\left(-{\frac {1}{4}}\right)!$ | (-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 |
| 1.20205690315959428539 | ثابت أبيري | | $\zeta (3)$ | $\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots =$ ${\frac {1}{2}}\sum _{n=1}^{\infty }{\frac {H_{n}}{n^{2}}}={\frac {1}{2}}\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }{\frac {1}{ij(i{+}j)}}=\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}{\frac {\mathrm {d} x\mathrm {d} y\mathrm {d} z}{1-xyz}}$ | Sum[n=1 to ∞] {1/n^3} | غ.ك | 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 |
| 0.91596559417721901505 | ثابت كاتالان | | ${C}$ | $\int _{0}^{1}\!\!\int _{0}^{1}\!\!{\frac {1}{1{+}x^{2}y^{2}}}\,dx\,dy=\!\sum _{n=0}^{\infty }\!{\frac {(-1)^{n}}{(2n{+}1)^{2}}}\!=\!{\frac {1}{1^{2}}}{-}{\frac {1}{3^{2}}}{+}{\cdots }$ | Sum[n=0 to ∞] {(-1)^n/(2n+1)^2} | م | A006752 | [0;1,10,1,8,1,88,4,1,1,7,22,1,2,3,26,1,11,...] | 1864 | 0.91596559417721901505460351493238411 |
| 0.78539816339744830961 | بيتا(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)} | م | 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 |
| 0.001317641154853178109 | ثابت روجر هيث براون | | ${C_{_{HBM}}}$ | ${\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}}}\right)^{7}\left(1+{\frac {7p_{n}+1}{p_{n}^{2}}}\right)}}$ | N[prod[n=1 to ∞] {((1-1/prime(n))^7) *(1+(7*prime(n)+1) /(prime(n)^2))}] | م | A118228 | [0;758,1,13,1,2,3,56,8,1,1,1,1,1,143,1,1,1,2,...] | | 0.00131764115485317810981735232251358 |
| 0.56755516330695782538 | الوحدة النمطية للرفع الوحدة التخيليةi | | $|^{\infty }{i}|$ | $\lim _{n\to \infty }\left|^{n}i\right|=\left|\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}\right|$ | 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 |
| 0.78343051071213440705 | حلم الطالب الجامعي (1) ليوهان بيرنولي | | ${I}_{1}$ | $\int _{0}^{1}\!x^{x}\,dx=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{n}}}={\frac {1}{1^{1}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}-{\cdots }$ | 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 |
| 1.291285997062663540407 | حلم الطالب الجامعي (2) ليوهان بيرنولي | | ${I}_{2}$ | $\int _{0}^{1}\!{\frac {1}{x^{x}}}\,dx=\sum _{n=1}^{\infty }{\frac {1}{n^{n}}}={\frac {1}{1^{1}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{4}}}+\cdots$ | 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 |
| 0.70523017179180096514 | ثابت بريموريال | | ${P_{\#}}$ | ${\underset {p_{n}:\,{prime}}{\sum _{n=1}^{\infty }{\frac {1}{p_{n}\#}}={\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{30}}+{\frac {1}{210}}+...=\sum _{k=1}^{\infty }\prod _{n=1}^{k}{\frac {1}{p_{n}}}}}$ | Sum[k=1 to ∞] (prod[n=1 to k] {1/prime(n)}) | غ.ك | A064648 | [0;1,2,2,1,1,4,1,2,1,1,6,13,1,4,1,16,6,1,1,4,...] | | 0.70523017179180096514743168288824851 |
| 0.14758361765043327417 | صيغة بيلي-بوروين-بلوف | | ${C}$ | ${\frac {1}{\pi }}\arctan {\frac {1}{2}}={\frac {1}{\pi }}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2^{2n+1})(2n+1)}}$ $={\frac {1}{\pi }}\left({\frac {1}{2}}-{\frac {1}{3\cdot 2^{3}}}+{\frac {1}{5\cdot 2^{5}}}-{\frac {1}{7\cdot 2^{7}}}+\cdots \right)$ | Arctan(1/2)/pi | م | 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 |
| 0.15915494309189533576 | ثابت بلوف | | ${A}$ | ${\frac {1}{2\pi }}$ | 1/(2 pi) | م | A086201 | [0;6,3,1,1,7,2,146,3,6,1,1,2,7,5,5,1,4,1,2,42,...] | | 0.15915494309189533576888376337251436 |
| 0.29156090403081878013 | ثابت ديمر ثنائي الأبعاد 2D, | | ${\frac {C}{\pi }}$ C= ثابت كاتالان | $\int \limits _{-\pi }^{\pi }{\frac {\cosh ^{-1}\left({\frac {\sqrt {\cos(t)+3}}{\sqrt {2}}}\right)}{4\pi }}\,dt$ | 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 |
| 0.498015668118356042 0.15494982830181068512 i | المضروب (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 | خ | 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 |
| 2.09455148154232659148 | ثابت واليس | | $W$ | ${\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}$ | (((45-sqrt(1929)) /18))^(1/3)+ (((45+sqrt(1929)) /18))^(1/3) | ج | 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 |
| 0.723648402298200009408 | ثابت سرناك | | ${C_{sa}}$ | $\prod _{p>2}{\Big1-{\frac {p+2}{p^{3}}}{\Big )}$ | N[prod[k=2 to ∞] {1-(prime(k)+2) /(prime(k)^3)}] | م | 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 |
| 0.632120558828557678404 | الثابت الزمني | | ${\tau }$ | $\lim _{n\to \infty }1-{\frac {!n}{n!}}=\lim _{n\to \infty }P(n)=\int _{0}^{1}e^{-x}dx=1{-}{\frac {1}{e}}=$ $\sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n!}}={\frac {1}{1!}}{-}{\frac {1}{2!}}{+}{\frac {1}{3!}}{-}{\frac {1}{4!}}{+}{\frac {1}{5!}}{-}{\frac {1}{6!}}{+}\cdots$ | lim_(n->∞) (1- !n/n!) !n=subfactorial | م | 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 |
| 1.04633506677050318098 | ثابت مينكوفسكي-سيجل | | $F_{1}$ | $\prod _{n=1}^{\infty }{\frac {n!}{{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}{\sqrt[{12}]{1+{\tfrac {1}{n}}}}}}$ | 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 |
| 5.244115108584239620929 | ثابت ليمنيسكيت | | $2\varpi$ | ${\frac {[\Gamma ({\tfrac {1}{4}})]^{2}}{\sqrt {2\pi }}}=4\int _{0}^{1}{\frac {dx}{\sqrt {(1-x^{2})(2-x^{2})}}}$ | 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 |
| 0.661707182267176235155 | ثابت روبين | | $\Delta (3)$ | ${\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}$ | (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 |
| 1.30357726903429639125 | ثابت كونواي | | ${\lambda }$ | ${\begin{smallmatrix}x^{71}\quad \ -x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\+x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\-12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\-10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}\quad \ -7x^{21}+9x^{20}\\+3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\+5x^{9}+x^{7}\quad \ -7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\quad \quad \quad \end{smallmatrix}}$ | | ج | A014715 | [1;3,3,2,2,54,5,2,1,16,1,30,1,1,1,2,2,1,14,1,...] | 1987 | 1.30357726903429639125709911215255189 |
| 1.18656911041562545282 | ثابت ليفي | | ${\beta }$ | ${\frac {\pi ^{2}}{12\,\ln 2}}$ | 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 |
| 0.83564884826472105333 | مبرهنة باكر | | $\beta _{3}$ | $\int _{0}^{1}{\frac {\mathrm {d} t}{1+t^{3}}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{3n+1}}={\frac {1}{3}}\left(\ln 2+{\frac {\pi }{\sqrt {3}}}\right)$ | 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 |
| 23.10344790942054161603 | متتالية كيمبنر(0) | | ${K_{0}}$ | $1{+}{\frac {1}{2}}{+}{\frac {1}{3}}{+}\cdots {+}{\frac {1}{9}}{+}{\frac {1}{11}}{+}\cdots {+}{\frac {1}{19}}{+}{\frac {1}{21}}{+}\cdots$ ${+}{\frac {1}{99}}{+}{\frac {1}{111}}{+}\cdots {+}{\frac {1}{119}}{+}{\frac {1}{121}}{+}\cdots$ | 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 |
| 0.989431273831146951741 | ثابت ليبسج | | ${C_{1}}$ | $\lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}{-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}\!\!\right)$ | 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 |
| 0.19452804946532511361 | المعامل الثاني لدي بو ريموند | | ${C_{2}}$ | ${\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1$ | (e^2-7)/2 | م | A062546 | [0;5,7,9,11,13,15,17,19,21,23,25,27,29,31,...] = [0;2p+3], p∈ℕ | | 0.19452804946532511361521373028750390 |
| 0.78853056591150896106 | ثابت لورث | | $C_{L}$ | $\sum _{n=2}^{\infty }{\frac {\ln \left({\frac {n}{n-1}}\right)}{n}}$ | 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 |
| 1.187452351126501054595 | ثابت غوياس $_α$ | | $F_{\alpha }$ | $x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots$ | | | 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 |
| 2.293166287411861031508 | ثابت غوياس $_β$ | | $F_{\beta }$ | $x^{x+1}=(x+1)^{x}$ | 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 |
| 0.82246703342411321823 | ثابت نيسلون-رامانجن | | ${\frac {{\zeta }(2)}{2}}$ | ${\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}{\frac {1}{5^{2}}}{-}\cdots$ | Sum[n=1 to ∞] {((-1)^(n+1))/n^2} | م | 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 |
| 0.69314718055994530941 | اللوغارتم الطبيعي للرقم 2 | | $Ln(2)$ | $\sum _{n=1}^{\infty }{\frac {1}{n2^{n}}}=\sum _{n=1}^{\infty }{\frac {({-}1)^{n+1}}{n}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\cdots }$ | Sum[n=1 to ∞] {(-1)^(n+1)/n} | م | 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 |
| 0.47494937998792065033 | ثابت ويرستراس | | $\sigma ({\tfrac {1}{2}})$ | ${\frac {e^{\frac {\pi }{8}}{\sqrt {\pi }}}{4\cdot 2^{3/4}{({\frac {1}{4}}!)^{2}}}}$ | (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 |
| 0.577215664901532860606 | ثابت أويلر-ماسكيروني | | ${\gamma }$ | $\sum _{n=1}^{\infty }\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2^{n}+k}}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln \left(1+{\frac {1}{n}}\right)\right)$ $=\int _{0}^{1}-\ln \left(\ln {\frac {1}{x}}\right)\,dx=-\Gamma '(1)=-\Psi (1)$ | 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 |
| 1.38135644451849779337 | ثابت بيتا كينسر ماهلر لمتعددة الحدود | | $\beta$ | $e^{^{\textstyle {\frac {2}{\pi }}\displaystyle {\int _{0}^{\frac {\pi }{3}}}\textstyle {t\tan t\ dt}}}=e^{^{\displaystyle {\,\int _{\frac {-1}{3}}^{\frac {1}{3}}}\textstyle {\,\ln \lfloor 1+e^{2\pi it}}\rfloor dt}}$ | 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 |
| 1.358456274182988435206 | الدوامة الذهبية | | $c$ | $\varphi ^{\frac {2}{\pi }}=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{\frac {2}{\pi }}$ | 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 |
| 0.57595996889294543964 | ثابت ستيفين | | $C_{S}$ | $\prod _{n=1}^{\infty }\left(1-{\frac {p}{p^{3}-1}}\right)$ | Prod[n=1 to ∞] {1-hprime(n) /(hprime(n)^3-1)} | م | A065478 | [0;1,1,2,1,3,1,3,1,2,1,77,2,1,1,10,2,1,1,1,7,...] | | 0.57595996889294543964316337549249669 |
| 0.73908513321516064165 | عدد دوتي | | $d$ | $\lim _{x\to \infty }\cos ^{[x]}(c)=\lim _{x\to \infty }\underbrace {\cos(\cos(\cos(\cdots (\cos(c)))))} _{x}$ | cos(c)=c | م | A003957 | [0;1,2,1,4,1,40,1,9,4,2,1,15,2,12,1,21,1,17,...] | | 0.73908513321516064165531208767387340 |
| 0.67823449191739197803 | ثابت تانيجوتشي | | $C_{T}$ | خطأ رياضيات (خطأ في الصياغة): {\displaystyle \prod_{n = 1}^\infty \left(1 - \frac{3}{{p_n}^3}+\frac{2}{{p_n}^4}+rac{1}{{p_n}^5}-rac{1}{{p_n}^6} ight) } $\scriptstyle p_{n}=\,ext{prime}$ | Prod[n=1 to ∞] {1 -3/ithprime(n)^3 +2/ithprime(n)^4 +1/ithprime(n)^5 -1/ithprime(n)^6} | م | A175639 | [0;1,2,9,3,1,2,9,11,1,13,2,15,1,1,1,2,4,1,1,1,...] | | 0.67823449191739197803553827948289481 |
| 1.85407467730137191843 | ثابت جاووس ليمنيسكيت | | $L{\text{/}}{\sqrt {2}}$ | $\int \limits _{0}^{\infty }{\frac {\mathrm {d} x}{\sqrt {1+x^{4}}}}={\frac {1}{4{\sqrt {\pi }}}}\,\Gamma \left({\frac {1}{4}}\right)^{2}={\frac {4\left({\frac {1}{4}}!\right)^{2}}{\sqrt {\pi }}}$ $\scriptstyle \Gamma{\text{= Gamma function}}$ | 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 |
| 1.75874362795118482469 | ثابت الضرب اللانهائي | | $Pr_{1}$ | $\prod _{n=2}^{\infty }{\Big1+{\frac {1}{n}}{\Big )}^{\frac {1}{n}}$ | 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 |
| 1.86002507922119030718 | حلزون تيودوروس | | $\partial$ | $\sum _{n=1}^{\infty }{\frac {1}{{\sqrt {n^{3}}}+{\sqrt {n}}}}=\sum _{n=1}^{\infty }{\frac {1}{{\sqrt {n}}(n+1)}}$ | 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 | متداخلة جذرية S$_5$ | | $S_{5}$ | $\displaystyle {\frac {{\sqrt {21}}+1}{2}}=\scriptstyle \,{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+\cdots }}}}}}}}}}\;$ $=1+\,\scriptstyle {\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-\cdots }}}}}}}}}}\;$ | (sqrt(21)+1)/2 | ج | 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 |
| 0.70710678118654752 br> +0.70710 67811 86547 524 i> | الجذر التربيعي للوحدة التخيليةi | | ${\sqrt {i}}$ | ${\sqrt[{4}]{-1}}={\frac {1+i}{\sqrt {2}}}=e^{\frac {i\pi }{4}}=\cos \left({\frac {\pi }{4}}\right)+i\sin \left({\frac {\pi }{4}}\right)$ | (1+i)/(sqrt 2) | ج خ | 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 |
| 0.809394020540639130717 | ثابت اللادي – جرينستيد | | ${{\mathcal {A}}_{AG}}$ | $e^{-1+\sum \limits _{k=2}^{\infty }\sum \limits _{n=1}^{\infty }{\frac {1}{nk^{n+1}}}}=e^{-1-\sum \limits _{k=2}^{\infty }{\frac {1}{k}}\ln \left(1-{\frac {1}{k}}\right)}$ | 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 |
| 2.58498175957925321706 | ثابت شيربينسكي | | ${K}$ | $\pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma ({\tfrac {1}{4}})^{4}}}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )$ $=\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4}})\right)$ | -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 |
| 1.73245471460063347358 | ثابت أويلر – ماتشيروني | | ${\frac {1}{\gamma }}$ | $\left(\int _{0}^{1}-\log \left(\log {\frac {1}{x}}\right)\,dx\right)^{-1}=\sum _{n=1}^{\infty }(-1)^{n}(-1+\gamma )^{n}$ | 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 |
| 1.435991124176917432355 | ثابت يبيسج | | ${L_{1}}$ | $\prod _{\begin{smallmatrix}i=0\\j\neq i\end{smallmatrix}}^{n}{\frac {x-x_{i}}{x_{j}-x_{i}}}={\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\lfloor \sin {\frac {3t}{2}}\rfloor }{\sin {\frac {t}{2}}}}\,dt={\frac {1}{3}}+{\frac {2{\sqrt {3}}}{\pi }}$ | 1/3 + 2*sqrt(3)/pi | م | 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 |
| 3.24697960371746706105 | الجذر الفضي | | $\varsigma$ | $2+2\cos {\frac {2\pi }{7}}=\textstyle 2+{\frac {2+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}{1+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}}$ | 2+2 cos(2Pi/7) | ج | A116425 | [3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...] | | 3.24697960371746706105000976800847962 |
| 1.94359643682075920505 | مؤشر أويلر | | $ET$ | ${\underset {p{\text{= primes}}}{\prod _{p}{\Big1+{\frac {1}{p(p-1)}}{\Big )}}}={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}={\frac {315\zeta (3)}{2\pi ^{4}}}$ | 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 |
| 1.495348781221220541911 | الجذر الرابع ل5 | | ${\sqrt[{4}]{5}}$ | ${\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,\cdots }}}}}}}}}}$ | (5(5(5(5(5(5(5) ^1/5)^1/5)^1/5) ^1/5)^1/5)^1/5) ^1/5 ... | ج | A011003 | [1;2,53,4,96,2,1,6,2,2,2,6,1,4,1,49,17,2,3,2,...] | | 1.49534878122122054191189899414091339 |
| 0.87228404106562797617 | مساحة دائرة فورد | | $A_{CF}$ | $\sum _{q\geq 1}\sum _{(p,q)=1 \atop 1\leq p<q}\pi \left({\frac {1}{2q^{2}}}\right)^{2}{\underset {\zeta{\text{= Riemann Zeta Function}}}{={\frac {\pi }{4}}{\frac {\zeta (3)}{\zeta (4)}}={\frac {45}{2}}{\frac {\zeta (3)}{\pi ^{3}}}}}$ | 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 |
| 1.08232323371113819151 | زيتا(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} | م | A013662 | [1;12,6,1,3,1,4,183,1,1,2,1,3,1,1,5,4,2,7,23,...] | ? | 1.08232323371113819151600369654116790 |
| 1.56155281280883027491 | عدد مثلثي مربعي للرقم 2. | | ${R_{2}}$ | ${\frac {{\sqrt {17}}-1}{2}}=\,\scriptstyle {\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+\cdots }}}}}}}}}}}}\,\,-1$ $=\,\scriptstyle {\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-\cdots }}}}}}}}}}}}\textstyle$ | (sqrt(17)-1)/2 | ج | 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 |
| 9.86960440108935861883 | مربع باي | | ${\pi }^{2}$ | $6\,\zeta (2)=6\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {6}{1^{2}}}+{\frac {6}{2^{2}}}+{\frac {6}{3^{2}}}+{\frac {6}{4^{2}}}+\cdots$ | 6 Sum[n=1 to ∞] {1/n^2} | م | A002388 | [9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,1,3,...] | | 9.86960440108935861883449099987615114 |
| 1.32471795724474602596 | العدد البلاستيكي | | ${\rho }$ | ${\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+\!{\sqrt {\frac {23}{108}}}}}+\!{\sqrt[{3}]{{\frac {1}{2}}-\!{\sqrt {\frac {23}{108}}}}}$ | (1+(1+(1+(1+(1+(1) ^(1/3))^(1/3))^(1/3)) ^(1/3))^(1/3))^(1/3) | ج | A060006 | [1;3,12,1,1,3,2,3,2,4,2,141,80,2,5,1,2,8,2,...] | 1929 | 1.32471795724474602596090885447809734 |
| 2.37313822083125090564 | ثابت ليفي$_2$ | | $2\,ln\,\gamma$ | ${\frac {\pi ^{2}}{6ln(2)}}$ | Pi^(2)/(6*ln(2)) | م | A174606 | [2;2,1,2,8,57,9,32,1,1,2,1,2,1,2,1,2,1,3,2,...] | 1936 | 2.37313822083125090564344595189447424 |
| 0.85073618820186726036 | متسلسلة طوي الورق | | ${P_{f}}$ | $\sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}$ | 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 |
| 1.1563626843322697168533 | ثابت تكرار المكعب | | ${\sigma _{3}}$ | $\prod _{n=1}^{\infty }n^{{3}^{-n}}={\sqrt[{3}]{1{\sqrt[{3}]{2{\sqrt[{3}]{3\cdots }}}}}}=1^{1/3}\;2^{1/9}\;3^{1/27}\cdots$ | 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 |
| 1.261859507142914874199 | البعد الكسري لمنحنى ندفة الثلج لكوخ | | ${C_{k}}$ | ${\frac {\log 4}{\log 3}}$ | log(4)/log(3) | م | A100831 | [1;3,1,4,1,1,11,1,46,1,5,112,1,1,1,1,1,3,1,7,...] | | 1.26185950714291487419905422868552171 |
| 6.58088599101792097085 | ثابت فورودا | | $2^{\,e}$ | $2^{e}$ | 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 | ثابت ميرتنز-ميسيل | | ${M}$ | $\lim _{n\rightarrow \infty }\!\!\left(\sum _{p\leq n}{\frac {1}{p}}\!-\ln(\ln(n))\!\right)\!\!={\underset {\!\!\!\!\gamma :\,{\text{Euler constant}},\,\,p:\,{\text{prime}}}{\!\gamma \!+\!\!\sum _{p}\!\left(\!\ln \!\left(\!1\!-\!{\frac {1}{p}}\!\right)\!\!+\!{\frac {1}{p}}\!\right)}}$ | gamma+ Sum[n=1 to ∞] {ln(1-1/prime(n)) +1/prime(n)} | م | 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 |
| 4.81047738096535165547 | ثابت جون | | $\gamma$ | ${\sqrt[{i}]{i}}=i^{-i}=(i^{i})^{-1}=(((i)^{i})^{i})^{i}=e^{\frac {\pi }{2}}={\sqrt {\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}}}$ | e^(π/2) | م | 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 ± 0.86602540378443 i | الجذر التكعيبي للرقم 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) | خ ج | 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 |
| 0.110001000000000000000001 | عدد ليوفيل نص صغير | | ${\text{£}}_{Li}$ | $\sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots$ | Sum[n=1 to ∞] {10^(-n!)} | م | A012245 | [1;9,1,999,10,9999999999999,1,9,999,1,9] | | 0.11000100000000000000000100... |
| 0.06598803584531253707 | النهاية الصغرى لرفع الأساس e بالأس e. | | ${e}^{-e}$ | $\left({\frac {1}{e}}\right)^{e}$ | 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 |
| 1.83928675521416113255 | ثابت تريبوناكسي | | ${\phi _}_{3}$ | $\textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}=\scriptstyle \,1+\left({\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+...}}}}}}\right)^{-1}$ | (1/3)*(1+(19+3 *sqrt(33))^(1/3) +(19-3 *sqrt(33))^(1/3)) | ج | A058265 | [1;1,5,4,2,305,1,8,2,1,4,6,14,3,1,13,5,1,7,...] | | 1.83928675521416113255185256465328660 |
| 0.366512920581664327012 | متوسط توزيع جامبل | | ${ll_{2}}$ | $-\ln(\ln(2))$ | -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 |
| 36.46215960720791177099 | باي مرفوع بالأس باي | | $\pi ^{\pi }$ | $\pi ^{\pi }$ | pi^pi | | A073233 | [36;2,6,9,2,1,2,5,1,1,6,2,1,291,1,38,50,1,2,...] | | 36.4621596072079117709908260226921236 |
| 0.53964549119041318711 | ثابت إيواتشيميسكو | | $2+\zeta ({\tfrac {1}{2}})$ | ${2{-}(1{+}{\sqrt {2}})\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{\sqrt {n}}}}=\gamma +\sum _{n=1}^{\infty }{\frac {(-1)^{2n}\;\gamma _{n}}{2^{n}n!}}$ | 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 |
| 15.1542622414792641897 | مجموعة الهروب | | $e^{e}$ | $\sum _{n=0}^{\infty }{\frac {e^{n}}{n!}}=\lim _{n\to \infty }\left({\frac {1+n}{n}}\right)^{n^{-n}(1+n)^{1+n}}$ | 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 |
| 0.64624543989481330426 | ثابت جرمين-ماصر | | ${C}$ | $\gamma {\beta }(1)\!+\!{\beta }'(1)\!=\pi \!\left(-\!\ln \Gamma ({\tfrac {1}{4}})+{\tfrac {3}{4}}\pi +{\tfrac {1}{2}}\ln 2+{\tfrac {1}{2}}\gamma \right)$ $=\pi \!\left(-\!\ln({\tfrac {1}{4}}!)+{\tfrac {3}{4}}\ln \pi -{\tfrac {3}{2}}\ln 2+{\tfrac {1}{2}}\,\gamma \right)$ $\scriptstyle \gamma ={\text{Euler–Mascheroni constant}}=0.5772156649\ldots$ $\scriptstyle \beta={\text{Beta function}},\quad \scriptstyle \Gamma={\text{Gamma function}}$ | 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 |
| 1.11072073453959156175 | النسبة بين مربع محاط بدائرة | | ${\frac {\pi }{2{\sqrt {2}}}}$ | $\sum _{n=1}^{\infty }{\frac {({-}1)^{\lfloor {\frac {n-1}{2}}\rfloor }}{2n+1}}={\frac {1}{1}}+{\frac {1}{3}}-{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}-{\cdots }$ | sum[n=1 to ∞] {(-1)^(floor( (n-1)/2)) /(2n-1)} | م | A093954 | [1;9,31,1,1,17,2,3,3,2,3,1,1,2,2,1,4,9,1,3,...] | | 1.11072073453959156175397024751517342 |
| 1.45607494858268967139 | ثابت باكهاوس | | ${B}$ | $\lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad \scriptstyle {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}$ $P(x)=\sum _{k=1}^{\infty }{\underset {p_{k}{\text{ prime}}}{p_{k}x^{k}}}=1+2x+3x^{2}+5x^{3}+\cdots$ | 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 |
| 1.85193705198246617036 | ثابت غيبس | | ${Si(\pi )}$ تكامل الجيب | $\int _{0}^{\pi }{\frac {\sin t}{t}}\,dt=\sum \limits _{n=1}^{\infty }(-1)^{n-1}{\frac {\pi ^{2n-1}}{(2n-1)(2n-1)!}}$ $=\pi -{\frac {\pi ^{3}}{3\cdot 3!}}+{\frac {\pi ^{5}}{5\cdot 5!}}-{\frac {\pi ^{7}}{7\cdot 7!}}+\cdots$ | SinIntegral[Pi] | | A036792 | [1;1,5,1,3,15,1,5,3,2,7,2,1,62,1,3,110,1,39,...] | | 1.85193705198246617036105337015799136 |
| 0.23571113171923293137 | ثابت كوبلاند – إيردوس | | ${{\mathcal {C}}_{CE}}$ | $\sum _{n=1}^{\infty }{\frac {p_{n}}{10^{n+\sum \limits _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor }}}$ | sum[n=1 to ∞] {prime(n) /(n+(10^ sum[k=1 to n]{floor (log_10 prime(k))}))} | غ.ك | A033308 | [0;4,4,8,16,18,5,1,1,1,1,7,1,1,6,2,9,58,1,3,...] | | 0.23571113171923293137414347535961677 |
| 1.523627086202492106277 | البعد الكسري لمنحني التنين | | ${C_{d}}$ | ${\frac {\log \left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87}}}}+{\sqrt[{3}]{73+6{\sqrt {87}}}}}{3}}\right)}{\log(2)}}$ | (log((1+(73-6 sqrt(87))^1/3+ (73+6 sqrt(87))^1/3)/3))/ log(2))) | م | | [1;1,1,10,12,2,1,149,1,1,1,3,11,1,3,17,4,1,...] | | 1.52362708620249210627768393595421662 |
| 1.78221397819136911177 | ثابت جروثينديك | | ${K_{R}}$ | ${\frac {\pi }{2\log(1+{\sqrt {2}})}}$ | 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 |
| 1.58496250072115618145 | بعد هاوسدورف، مثلث سيربنسكي | | ${log_{2}3}$ | ${\frac {\log 3}{\log 2}}={\frac {\sum _{n=0}^{\infty }{\frac {1}{2^{2n+1}(2n+1)}}}{\sum _{n=0}^{\infty }{\frac {1}{3^{2n+1}(2n+1)}}}}={\frac {{\frac {1}{2}}+{\frac {1}{24}}+{\frac {1}{160}}+\cdots }{{\frac {1}{3}}+{\frac {1}{81}}+{\frac {1}{1215}}+\cdots }}$ | ( Sum[n=0 to ∞] {1/ (2^(2n+1) (2n+1))})/ (Sum[n=0 to ∞] {1/ (3^(2n+1) (2n+1))}) | م | A020857 | [1;1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...] | | 1.58496250072115618145373894394781651 |
| 1.30637788386308069 | ثابت ميلز | | ${\theta }$ | $\lfloor \theta ^{3^{n}}\rfloor$ 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 |
| 2.02988321281930725004 | عقدة الرقم 8 | | ${V_{8}}$ | $2{\sqrt {3}}\,\sum _{n=1}^{\infty }{\frac {1}{n{2n \choose n}}}\sum _{k=n}^{2n-1}{\frac {1}{k}}=6\int \limits _{0}^{\pi /3}\log \left({\frac {1}{2\sin t}}\right)\,dt=$ $\scriptstyle {\frac {\sqrt {3}}{9}}\,\sum \limits _{n=0}^{\infty }{\frac {(-1)^{n}}{27^{n}}}\,\left\{\!{\frac {18}{(6n+1)^{2}}}-{\frac {18}{(6n+2)^{2}}}-{\frac {24}{(6n+3)^{2}}}-{\frac {6}{(6n+4)^{2}}}+{\frac {2}{(6n+5)^{2}}}\!\right\}$ | 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 |
| 262537412640768743.999999999999250073 | ثابت هيرميت-رامانوجان | | ${R}$ | $e^{\pi {\sqrt {163}}}$ | e^(π sqrt(163)) | م | A060295 | [262537412640768743;1,1333462407511,1,8,1,1,5,...] | 1859 | 262537412640768743.999999999999250073 |
| 1.74540566240734686349 | المتوسط التوافقي خنشن | | ${K_{-1}}$ | ${\frac {\log 2}{\sum \limits _{n=1}^{\infty }{\frac {1}{n}}\log {\bigl1{+}{\frac {1}{n(n+2)}}{\bigr )}}}=\lim _{n\to \infty }{\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}$ a$_1$ ... a$_n$ هي عناصر كسر مستمر [a$_0$; a$_1$, a$_2$, ..., a$_n$] | (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 |
| 1.648721270700128146848 | الجذر التربيعي للعدد ه | | ${\sqrt {e}}$ | $\sum _{n=0}^{\infty }{\frac {1}{2^{n}n!}}=\sum _{n=0}^{\infty }{\frac {1}{(2n)!!}}={\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{48}}+\cdots$ | Sum[n=0 to ∞] {1/(2^n n!)} | م | 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 |
| 1.017343061984449139714 | زيتا(6) | | $\zeta (6)$ | ${\frac {\pi ^{6}}{945}}\!=\!\prod _{n=1}^{\infty }\!{\underset {p_{n}:{\text{ prime}}}{\frac {1}{{1-p_{n}}^{-6}}}}={\frac {1}{1\!-\!2^{-6}}}\!\cdot \!{\frac {1}{1\!-\!3^{-6}}}\!\cdot \!{\frac {1}{1\!-\!5^{-6}}}\cdots$ | Prod[n=1 to ∞] {1/(1-ithprime (n)^-6)} | م | A013664 | [1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...] | | 1.01734306198444913971451792979092052 |
| 0.108410151223111361511 | ثابت تروت | | $\mathrm {T} _{1}$ | $\textstyle [1,0,8,4,1,0,1,5,1,2,2,3,1,1,1,3,6,...]$ ${\tfrac {1}{1+{\tfrac {1}{0+{\tfrac {1}{8+{\tfrac {1}{4+{\tfrac {1}{1+{\tfrac {1}{0+1{/\cdots }}}}}}}}}}}}}$ | | | A039662 | [0;9,4,2,5,1,2,2,3,1,1,1,3,6,1,5,1,1,2,...] | | 0.10841015122311136151129081140641509 |
| 0.0078749969978123844 | ثابت شاتان | | $\Omega$ | $\sum _{p\in P}2^{-|p|}$ | | م | A100264 | [0; 126, 1, 62, 5, 5, 3, 3, 21, 1, 4, 1] | 1975 | 0.0078749969978123844 |
| 0.83462684167407318628 | ثابت جاووس نص صغير | | ${G}$ | ${\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {4{\sqrt {2}}\,({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}$ | (4 sqrt(2)((1/4)!)^2) /pi^(3/2) | م | A014549 | [0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...] | | 0.83462684167407318628142973279904680 |
| 1.451369234883381050283 | ثابت سولدنر رامانجن | | ${\mu }$ | $\mathrm {li} (x)=\int \limits _{0}^{x}{\frac {dt}{\ln t}}=0{\color {White}{......}}$ li = لوغارتم خطي $\mathrm {li} (x)\;=\;\mathrm {Ei} (\ln {x}){\color {White}{........}}$ Ei = تكامل أسي | FindRoot[li(x) = 0] | غ.ك | 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 |
| 0.64341054628833802618 | ثابت الكاهن | | $\xi _{2}$ | $\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}{\,\pm \cdots }$ $\,\,S_{0}=\,2,\,\,S_{k}=\,1+\prod \limits _{n=0}^{k-1}S_{n}{\text{ for}}\;k>0$ | | م | A080130 | [0; 1, 1, 1, 4, 9, 196, 16641, 639988804, ...] | 1891 | 0.64341054628833802618225430775756476 |
| 1.414213562373095048801 | الجذر التربيعي ل 2، ثابت فيثاغورس . | | ${\sqrt {2}}$ | $\!\prod _{n=1}^{\infty }\!\left(1\!+\!{\frac {(-1)^{n+1}}{2n-1}}\right)\!=\!\left(1\!+\!{\frac {1}{1}}\right)\!\left(1\!-\!{\frac {1}{3}}\right)\!\left(1\!+\!{\frac {1}{5}}\right)\cdots$ | prod[n=1 to ∞] {1+(-1)^(n+1) /(2n-1)} | ج | 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 |
| 1.77245385090551602729 | ثابت كارلسون ليفين | | ${\Gamma }({\tfrac {1}{2}})$ | ${\sqrt {\pi }}=\left(-{\frac {1}{2}}\right)!=\int _{-\infty }^{\infty }{\frac {1}{e^{x^{2}}}}\,dx=\int _{0}^{1}{\frac {1}{\sqrt {-\ln x}}}\,dx$ | sqrt (pi) | م | A002161 | [1;1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,...] | | 1.77245385090551602729816748334114518 |
| 1.05946309435929526456 | الفاصل الموسيقي بين نصف كل نغمة | | ${\sqrt[{12}]{2}}$ | ${\begin{array}{l|ccccccccccccr}\!2^{\frac {x}{12}}\!&\!\!\scriptstyle {0}&\!\!\!\!\scriptstyle {1}&\!\!\scriptstyle {2}&\!\!\scriptstyle {3}&\!\!\scriptstyle {4}&\scriptstyle {5}&\!\!\scriptstyle {6}&\!\!\scriptstyle {7}&\!\!\scriptstyle {8}&\!\!\scriptstyle {9}&\!\!\scriptstyle {10}&\!\!\scriptstyle {11}&\!\!\scriptstyle {12}\\\hline \!\scriptstyle {\textrm {Key}}\!&\!\scriptstyle {\mathrm {C_{1}} }&\!\!\scriptstyle {\mathrm {C^{\#}} }&\!\!\scriptstyle {\mathrm {D} }&\!\scriptstyle {\mathrm {D^{\#}} }&\!\!\scriptstyle {\mathrm {E} }&\scriptstyle {\mathrm {F} }&\!\scriptstyle {\mathrm {F^{\#}} }&\!\!\scriptstyle {\mathrm {G} }&\!\scriptstyle {\mathrm {G^{\#}} }&\!\!\scriptstyle {\mathrm {A} }&\!\scriptstyle {\mathrm {A^{\#}} }&\!\!\scriptstyle {\mathrm {B} }&\!\scriptstyle {\mathrm {C_{2}} }\end{array}}$(A = 440 Hz) | 2^(1/12) | ج | A010774 | [1;16,1,4,2,7,1,1,2,2,7,4,1,2,1,60,1,3,1,2,...] | | 1.05946309435929526456182529494634170 |
| 1.01494160640965362502 | ثابت جيسكنج | | ${\pi \ln \beta }$ | ${\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=$ $\textstyle {\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}+{\frac {1}{10^{2}}}\pm \cdots \right)$. | 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 |
| 2.62205755429211981046 | ثابت ليمنيسكاتي | | ${\varpi }$ | $\pi \,{G}=4{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {5}{4}}\right)^{2}}={\tfrac {1}{4}}{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {1}{4}}\right)^{2}}=4{\sqrt {\tfrac {2}{\pi }}}\left({\tfrac {1}{4}}!\right)^{2}$ | 4 sqrt(2/pi) ((1/4)!)^2 | م | 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 |
| 1.28242712910062263687 | ثابت جلايشر كين كيلن | | ${A}$ | $e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}$ | e^(1/12-zeta´{-1}) | م | A074962 | [1;3,1,1,5,1,1,1,3,12,4,1,271,1,1,2,7,1,35,...] | | 1.28242712910062263687534256886979172 |
| 4.227453533376265408- | دالة دي جاما (1/4) | | ${\psi }({\tfrac {1}{4}})$ | $-\gamma -{\frac {\pi }{2}}-3\ln {2}=-\gamma +\sum _{n=0}^{\infty }\left({\frac {1}{n+1}}-{\frac {1}{n+{\tfrac {1}{4}}}}\right)$ | -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 |
| 0.286747428434478734107 | ثابت الإهمال القوي | | $K_{2}$ | $\prod _{n=1}^{\infty }{\underset {p_{n}:{\text{ prime}}}{\left(1-{\frac {3p_{n}-2}{{p_{n}}^{3}}}\right)}}={\frac {6}{\pi ^{2}}}\prod _{n=1}^{\infty }{\underset {p_{n}:{\text{ prime}}}{\left(1-{\frac {1}{p_{n}(p_{n}+1)}}\right)}}$ | 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 |
| 3.62560990822190831193 | جاما(1/4) | | $\Gamma ({\tfrac {1}{4}})$ | $4\left({\frac {1}{4}}\right)!=\left(-{\frac {3}{4}}\right)!$ | 4(1/4)! | م | 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 |
| 1.66168794963359412129 | ثابت سموس | | ${\sigma }$ | $\prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots$ | prod[n=1 to ∞] {n ^(1/2)^n} | م | A065481 | [1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...] | | 1.66168794963359412129581892274995074 |
| 0.955316618124509278163 | الزاوية السحرية | | ${\theta _{m}}$ | $\arctan \left({\sqrt {2}}\right)=\arccos \left({\sqrt {\tfrac {1}{3}}}\right)\approx \textstyle {54.7356}^{\circ }$ | arctan(sqrt(2)) | م | A195696 | [0;1,21,2,1,1,1,2,1,2,2,4,1,2,9,1,2,1,1,1,3,...] | | 0.95531661812450927816385710251575775 |
| 1.78107241799019798523 | دالة بارنس | | $e^{\gamma }$ | $\prod _{n=1}^{\infty }{\frac {e^{\frac {1}{n}}}{1+{\tfrac {1}{n}}}}=\prod _{n=0}^{\infty }\left(\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac {1}{n+1}}=$ $\textstyle \left({\frac {2}{1}}\right)^{1/2}\left({\frac {2^{2}}{1\cdot 3}}\right)^{1/3}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{1/4}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{1/5}\cdots$ | 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 |
| 0.74759792025341143517 | ثابت رينيه لركن السيارات | | ${m}$ | $\int \limits _{0}^{\infty }exp\left(\!-2\int \limits _{0}^{x}{\frac {1-e^{-y}}{y}}dy\right)\!dx={e^{-2\gamma }}\int \limits _{0}^{\infty }{\frac {e^{-2\Gamma (0,n)}}{n^{2}}}$ | [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 |
| 1.273239544735162686151 | سلسلة رامانوجان-فورسيث | | ${\frac {4}{\pi }}$ | $\displaystyle \sum \limits _{n=0}^{\infty }\textstyle \left({\frac {(2n-3)!!}{(2n)!!}}\right)^{2}={1\!+\!\left({\frac {1}{2}}\right)^{2}\!+\!\left({\frac {1}{2\cdot 4}}\right)^{2}\!+\!\left({\frac {1\cdot 3}{2\cdot 4\cdot 6}}\right)^{2}+\cdots }$ | Sum[n=0 to ∞] {[(2n-3)!! /(2n)!!]^2} | غ.ك | A088538 | [1;3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...] | | 1.27323954473516268615107010698011489 |
| 1.444667861009766133658 | عدد ستينر، ه جذر ه | | ${\sqrt[{e}]{e}}$ | $e^{\frac {1}{e}}{\color {White}{...........}}$ | e^(1/e) | م | A073229 | [1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...] | | 1.44466786100976613365833910859643022 |
| 0.692200627555346353865 | الحد الأدنى للدالة ƒ(x) = x$^x$ | | ${\left({\frac {1}{e}}\right)}^{\frac {1}{e}}$ | ${e}^{-{\frac {1}{e}}}{\color {White}{..........}}$ = مقلوب عدد ستينر | 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 |
| 0.34053732955099914282 | ثابت السير العشوائي | | ${p(3)}$ | $1-\!\!\left({3 \over (2\pi )^{3}}\int \limits _{-\pi }^{\pi }\int \limits _{-\pi }^{\pi }\int \limits _{-\pi }^{\pi }{dx\,dy\,dz \over 3-\!\cos x-\!\cos y-\!\cos z}\right)^{\!-1}$ $=1-16{\sqrt {\tfrac {2}{3}}}\;\pi ^{3}\left(\Gamma ({\tfrac {1}{24}})\Gamma ({\tfrac {5}{24}})\Gamma ({\tfrac {7}{24}})\Gamma ({\tfrac {11}{24}})\right)^{-1}$ | 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 |
| 0.543258965342976706952 | نظرية بلوتش (المتغيرات المركبة) | | ${L}$ | $={\frac {\Gamma ({\tfrac {1}{3}})\;\Gamma ({\tfrac {5}{6}})}{\Gamma ({\tfrac {1}{6}})}}={\frac {(-{\tfrac {2}{3}})!\;(-1+{\tfrac {5}{6}})!}{(-1+{\tfrac {1}{6}})!}}$ | 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 |
| 0.187859642462067120248 | ثابت إم أر بي (مارفن راي بيرنز) | | $C_{_{MRB}}$ | $\sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots$ | 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 |
| 1.4670780794339754728977 | ثابت بورتر | | ${C}$ | ${\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}$ $\scriptstyle \gamma \,{\text{= Euler–Mascheroni Constant}}=0.5772156649\ldots$ $\scriptstyle \zeta '(2)\,{\text{= Derivative of }}\zeta (2)=-\sum \limits _{n=2}^{\infty }{\frac {\ln n}{n^{2}}}=-0.9375482543\ldots$ | 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 |
| 4.66920160910299067185 | ثابت فايينبوم δ | | ${\delta }$ | $\lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}\qquad \scriptstyle x\in (3.8284;\,3.8495)$ $\scriptstyle x_{n+1}=\,ax_{n}(1-x_{n})\quad {\text{or}}\quad x_{n+1}=\,a\sin(x_{n})$ | | م | A006890 | [4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...] | 1975 | 4.66920160910299067185320382046620161 |
| 2.50290787509589282228 | ثابت فايينبوم α | | $\alpha$ | $\lim _{n\to \infty }{\frac {d_{n}}{d_{n+1}}}$ | | م | A006891 | [2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...] | 1979 | 2.50290787509589282228390287321821578 |
| 0.62432998854355087099 | ثابت غولومب-ديكمان | | ${\lambda }$ | $\int \limits _{0}^{\infty }{\underset {{\text{Para }}x>2}{{\frac {f(x)}{x^{2}}}\,dx}}=\int \limits _{0}^{1}e^{\operatorname {Li} (n)}dn\quad \scriptstyle {\text{Li: Logarithmic integral}}$ | 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 |
| 23.1406926327792690057 | ثابت غيلفوند | | ${e}^{\pi }$ | $(-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}={\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2!}}+{\frac {\pi ^{3}}{3!}}+\cdots$ | Sum[n=0 to ∞] {(pi^n)/n!} | م | A039661 | [23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...] | | 23.1406926327792690057290863679485474 |
| 7.38905609893065022723 | الثابت المخروطى، ثابت شوارتزشيلد | | $e^{2}$ | $\sum _{n=0}^{\infty }{\frac {2^{n}}{n!}}=1+2+{\frac {2^{2}}{2!}}+{\frac {2^{3}}{3!}}+{\frac {2^{4}}{4!}}+{\frac {2^{5}}{5!}}+\cdots$ | Sum[n=0 to ∞] {2^n/n!} | م | 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 |
| 0.35323637185499598454 | ثابت هافنر – سارنك – مككورليي (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 |
| 0.60792710185402662866 | ثابت هافنر – سارنك – مككورليي (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) | م | A059956 | [0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...] | | 0.60792710185402662866327677925836583 |
| 0.12345678910111213141 | ثابت تشامبيرنوون | | $C_{10}$ | $\sum _{n=1}^{\infty }\;\sum _{k=10^{n-1}}^{10^{n}-1}{\frac {k}{10^{kn-9\sum _{j=0}^{n-1}10^{j}(n-j-1)}}}$ | | م | A033307 | [0;8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15,...] | 1933 | 0.12345678910111213141516171819202123 |
| 0.76422365358922066299 | ثابت رامانجن-لاندو | | $K$ | ${\frac {1}{\sqrt {2}}}\prod _{p\equiv 3\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}}\!\!={\frac {\pi }{4}}\prod _{p\equiv 1\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}}$ | | م | A064533 | [0;1,3,4,6,1,15,1,2,2,3,1,23,3,1,1,3,1,1,6,4,...] | | 0.76422365358922066299069873125009232 |
| 2.71828182845904523536 | العدد ه، العدد النيبيري، عدد أويلر | | ${e}$ | $\!\lim _{n\to \infty }\!\left(\!1\!+\!{\frac {1}{n}}\right)^{n}\!=\!\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\textstyle \cdots$ | Sum[n=0 to ∞] {1/n!} (* lim_(n->∞) (1+1/n)^n *) | م | 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 |
| 0.3678794411714423215955 | معكوس العدد ه، معكوس العدد النيبيري، معكوس عدد أويلر | | ${\frac {1}{e}}$ | $\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}={\frac {1}{0!}}-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+{\frac {1}{4!}}-{\frac {1}{5!}}+\cdots$ | Sum[n=2 to ∞] {(-1)^n/n!} | م | 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 |
| 0.69034712611496431946 | الحد الأعلى للدالة الأسية المكررة | | ${H}_{2n+1}$ | $\lim _{n\to \infty }{H}_{2n+1}=\textstyle \left({\frac {1}{2}}\right)^{\left({\frac {1}{3}}\right)^{\left({\frac {1}{4}}\right)^{\cdot ^{\cdot ^{\left({\frac {1}{2n+1}}\right)}}}}}={2}^{-3^{-4^{\cdot ^{\cdot ^{-2n-1}}}}}$ | 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 |
| 0.6583655992 | الحد الأدنى للدالة الأسية المكررة | | ${H}_{2n}$ | $\lim _{n\to \infty }{H}_{2n}=\textstyle \left({\frac {1}{2}}\right)^{\left({\frac {1}{3}}\right)^{\left({\frac {1}{4}}\right)^{\cdot ^{\cdot ^{\left({\frac {1}{2n}}\right)}}}}}={2}^{-3^{-4^{\cdot ^{\cdot ^{-2n}}}}}$ | 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. |
| 3.14159265358979323846264 | ط، ثابت أرخميدس، ثابت الدائرة، باي | | $\pi$ | $\lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}} _{n}$ | Sum[n=0 to ∞] {(-1)^n 4/(2n+1)} | م | A000796 | [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...] | | 3.14159265358979323846264338327950288 |
| 1.9287800 | ثابت رايت | | ${\omega }$ | $\left\lfloor 2^{2^{2^{\cdot ^{\cdot ^{2^{\omega }}}}}}\!\right\rfloor \scriptstyle {\text{= primes:}}\displaystyle \left\lfloor 2^{\omega }\right\rfloor \scriptstyle {\text{=3,}}\displaystyle \left\lfloor 2^{2^{\omega }}\right\rfloor \scriptstyle {\text{=13,}}\displaystyle \left\lfloor 2^{2^{2^{\omega }}}\right\rfloor \scriptstyle =16381,\ldots$ | | | A086238 | [1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3] | | 1.9287800 |
| 0.4636476090008061162142 | سلسلة ماشين-غريغوري | | $\arctan {\frac {1}{2}}$ | ${\underset {{\text{For }}x=1/2\qquad \qquad }{\sum _{n=0}^{\infty }{\frac {(\!-1\!)^{n}\,x^{2n+1}}{2n+1}}={\frac {1}{2}}{-}{\frac {1}{3\!\cdot \!2^{3}}}{+}{\frac {1}{5\!\cdot \!2^{5}}}{-}{\frac {1}{7\!\cdot \!2^{7}}}{+}\cdots }}$ | Sum[n=0 to ∞] {(-1)^n (1/2)^(2n+1) /(2n+1)} | غ.ك | A073000 | [0;2,6,2,1,1,1,6,1,2,1,1,2,10,1,2,1,2,1,1,1,...] | | 0.46364760900080611621425623146121440 |
| 0.6977746579640079820067 | ثابت الكسر المستمر، دالة بيسل | | ${C}_{CF}$ | ${\frac {I_{1}(2)}{I_{0}(2)}}={\frac {\sum \limits _{n=0}^{\infty }{\frac {n}{n!n!}}}{\sum \limits _{n=0}^{\infty }{\frac {1}{n!n!}}}}=\textstyle {\tfrac {1}{1+{\tfrac {1}{2+{\tfrac {1}{3+{\tfrac {1}{4+{\tfrac {1}{5+{\tfrac {1}{6+1{/\cdots }}}}}}}}}}}}}$ | (Sum [n=0 to ∞] {n/(n!n!)}) / (Sum [n=0 to ∞] {1/(n!n!)}) | غ.ك | 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 |
| 1.902160583104 | مبرهنة برون = Σ مجموع مقلوب الأعداد الأولية التوأم | | ${B}_{\,2}$ | $\textstyle {\underset {p,\,p+2:{\text{ prime}}}{\sum ({\frac {1}{p}}+{\frac {1}{p+2}})}}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots$ | | | A065421 | [1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2] | | 1.902160583104 |
| 0.870588379975 | مبرهنة برون = Σ مجموع مقلوب مجموعة التوأم الرباعي | | ${B}_{\,4}$ | $\textstyle {\sum ({\frac {1}{p}}+{\frac {1}{p+2}}+{\frac {1}{p+6}}+{\frac {1}{p+8}})}\scriptstyle \quad {p,\;p+2,\;p+6,\;p+8:{\text{ prime}}}$ $\textstyle {\left({\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+{\tfrac {1}{13}}\right)}+\left({\tfrac {1}{11}}+{\tfrac {1}{13}}+{\tfrac {1}{17}}+{\tfrac {1}{19}}\right)+\dots$ | | | A213007 | [0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1] | | 0.870588379975 |
| 0.63661977236758134307 | ثابت بوفون | | ${\frac {2}{\pi }}$ | ${\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots$ صيغة فييت | 2/Pi | م | 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 |
| 0.59634736232319407434 | ثابت جومبرتز | | ${G}$ | $\!\int \limits _{0}^{\infty }\!\!{\frac {e^{-n}}{1{+}n}}\,dn=\!\!\int \limits _{0}^{1}\!\!{\frac {1}{1{-}\ln n}}\,dn=\textstyle {\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}$ | integral[0 to ∞] {(e^-n)/(1+n)} | غ.ك | A073003 | [0;1,1,2,10,1,1,4,2,2,13,2,4,1,32,4,8,1,1,1,...] | | 0.59634736232319407434107849936927937 |
| ت | وحدة تخيلية | | ${i}$ | ${\sqrt {-1}}={\frac {\ln(-1)}{\pi }}\qquad \qquad \mathrm {e} ^{i\,\pi }=-1$ | sqrt(-1) | غ.ك، خ | | | 1501 to 1576 | i |
| 2.74723 82749 32304 33305 | ثابت رامانجن للمتداخلة الجذرية | | $R_{5}$ | $\scriptstyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}\;=\textstyle {\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}$ | (2+sqrt(5) +sqrt(15 -6 sqrt(5)))/2 | ج | | [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 | ثابت أوميجا | | ${\Omega }$ | $\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}=\,\left({\frac {1}{e}}\right)^{\left({\frac {1}{e}}\right)^{\cdot ^{\cdot ^{\left({\frac {1}{e}}\right)}}}}=e^{-\Omega }=e^{-e^{-e^{\cdot ^{\cdot ^{-e}}}}}$ | Sum[n=1 to ∞] {(-n)^(n-1)/n!} | م | A030178 | [0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,2,1,...] | | 0.56714329040978387299996866221035555 |
| 0.968946146259369380483 | بيتا(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} | م | A153071 | [0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...] | | 0.96894614625936938048363484584691860 |
| 2.236067977499789696409 | الجذر التربيعي ل 5، مجموع غاوس | | ${\sqrt {5}}$ | $\scriptstyle (n=5)\displaystyle \sum _{k=0}^{n-1}e^{\frac {2k^{2}\pi i}{n}}=1+e^{\frac {2\pi i}{5}}+e^{\frac {8\pi i}{5}}+e^{\frac {18\pi i}{5}}+e^{\frac {32\pi i}{5}}$ | Sum[k=0 to 4] {e^(2k^2 pi i/5)} | ج | 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 |
| 3.35988566624317755317 | ثابت فيبوناتشي | | $\Psi$ | $\sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots$ F$_n$: متتالية فيبوناتشي | Sum[n=1 to ∞] {1/Fibonacci[n]} | غ.ك | A079586 | [3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...] | | 3.35988566624317755317201130291892717 |
| 2.685452001065306445309 | ثابت خينتشين | | $K_{\,0}$ | $\prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\ln n/\ln 2}$ | Prod[n=1 to ∞] {(1+1/(n(n+2))) ^(ln(n)/ln(2))} | م | A002210 | [2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...] | 1934 | 2.68545200106530644530971483548179569 |