\(113\mathbf{\color{blue}{\;=\;}}56+57\) (som van opeenvolgende gehele getallen)

\(113\mathbf{\color{blue}{\;=\;}}49+64\) (som van opeenvolgende kwadraatgetallen)

\(113\mathbf{\color{blue}{\;=\;}}((0;0;7;8)\,(0;2;3;10)\,(0;4;4;9)\,(2;3;6;8)\,(4;5;6;6))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#5\}\)

\(113\mathbf{\color{blue}{\;=\;}}((0;0;2;2;2;2;3;3;3)\,(0;1;2;2;2;2;2;2;4)\,(1;1;1;1;1;3;3;3;3))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#3\}\)

\(113\mathbf{\color{blue}{\;=\;}}2^4+2^4+3^4\)

\(113\mathbf{\color{blue}{\;=\;}}2^5+[3^4][9^2]\mathbf{\color{blue}{\;=\;}}[2^6][4^3][8^2]+7^2\mathbf{\color{blue}{\;=\;}}5^4-[2^9][8^3]\mathbf{\color{blue}{\;=\;}}11^2-2^3\mathbf{\color{blue}{\;=\;}}25^2-[2^9][8^3]\mathbf{\color{blue}{\;=\;}}38^2-11^3\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[2px,border:1px brown dashed]{57^2-56^2}\mathbf{\color{blue}{\;=\;}}133^2-26^3\mathbf{\color{blue}{\;=\;}}8669^2-422^3\)

113.1

\(113\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~\)References Sum of Three Cubes

\(\qquad~~~~\)Getallen van de vorm \(~9m+4~\) of \(~9m+5~\) kunnen nooit als som van drie derdemachten geschreven worden.

\(\qquad~~~~\)In dit geval is \(m=12~~(+5)\).

\(113\mathbf{\color{blue}{\;=\;}}\)(som van vier derdemachten)

\(\qquad~~~~(z\gt1000)\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{2^3+(-4)^3+(-7)^3+8^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{5^3+(-7)^3+(-10)^3+11^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{23^3+(-25)^3+(-34)^3+35^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{38^3+(-40)^3+(-55)^3+56^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-7)^3+50^3+68^3+(-76)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{14^3+44^3+74^3+(-79)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{35^3+71^3+80^3+(-97)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{86^3+(-91)^3+(-97)^3+101^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{56^3+74^3+80^3+(-103)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-10)^3+(-73)^3+(-109)^3+119^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{11^3+44^3+119^3+(-121)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-46)^3+107^3+107^3+(-133)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{50^3+65^3+125^3+(-133)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-31)^3+(-64)^3+(-154)^3+158^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{137^3+(-142)^3+(-154)^3+158^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-16)^3+74^3+161^3+(-166)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-94)^3+(-112)^3+(-160)^3+185^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{35^3+(-127)^3+(-178)^3+197^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-28)^3+(-46)^3+(-199)^3+200^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{140^3+(-142)^3+(-199)^3+200^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-79)^3+161^3+191^3+(-220)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-55)^3+119^3+209^3+(-220)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{59^3+(-175)^3+(-178)^3+221^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{128^3+(-217)^3+(-265)^3+299^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-124)^3+(-175)^3+(-280)^3+308^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{227^3+(-229)^3+(-322)^3+323^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{113^3+(-244)^3+(-310)^3+350^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-103)^3+242^3+353^3+(-385)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-7)^3+(-190)^3+(-370)^3+386^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-103)^3+236^3+380^3+(-406)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-106)^3+329^3+338^3+(-418)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-76)^3+365^3+371^3+(-463)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-124)^3+(-154)^3+(-511)^3+518^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-205)^3+(-250)^3+(-595)^3+617^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{140^3+(-220)^3+(-610)^3+617^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-241)^3+(-475)^3+(-574)^3+677^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-106)^3+161^3+704^3+(-706)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-220)^3+458^3+665^3+(-724)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-448)^3+548^3+704^3+(-751)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-7)^3+152^3+764^3+(-766)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-91)^3+(-325)^3+(-775)^3+794^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{53^3+500^3+749^3+(-817)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{443^3+(-562)^3+(-787)^3+833^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{302^3+(-466)^3+(-796)^3+833^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-85)^3+380^3+821^3+(-847)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{350^3+380^3+809^3+(-856)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{185^3+(-346)^3+(-847)^3+863^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-151)^3+(-178)^3+(-868)^3+872^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-133)^3+245^3+905^3+(-910)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{422^3+(-661)^3+(-823)^3+917^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{17^3+(-679)^3+(-796)^3+935^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{473^3+479^3+854^3+(-943)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-70)^3+179^3+947^3+(-949)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{35^3+617^3+860^3+(-955)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{14^3+(-142)^3+(-976)^3+977^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-304)^3+497^3+977^3+(-1009)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{320^3+788^3+806^3+(-1015)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{680^3+(-868)^3+(-934)^3+1049^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{14^3+620^3+977^3+(-1054)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-328)^3+629^3+989^3+(-1057)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-670)^3+(-742)^3+(-823)^3+1082^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-565)^3+(-832)^3+(-847)^3+1109^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~(z\gt1000)\)

\(113\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

\(\qquad~~~~\)(fully searched up to \(z=1000)\)

\(\qquad~~~~\bbox[lightyellow,3px,border:1px blue solid]{~oplossing~onbekend~}\mathbf{\color{blue}{\;=\;}}\)

113.2

\(113^2\mathbf{\color{blue}{\;=\;}}15^2+112^2\mathbf{\color{blue}{\;=\;}}6385^2-6384^2\)

\(113^3\mathbf{\color{blue}{\;=\;}}664^2+1001^2\mathbf{\color{blue}{\;=\;}}791^2+904^2\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{6441^2-6328^2}\)

113.3

\(113^2\mathbf{\color{blue}{\;=\;}}12769\) en het omgekeerde \(96721\mathbf{\color{blue}{\;=\;}}311^2\). Hetzelfde doet zich voor bij

113.4

\(113^2\mathbf{\color{blue}{\;=\;}}10^2+112^2+5^3\)

113.5
  EEN PUZZEL  

\(\bbox[3px,border:1px solid blue]{\;Opgave\;}\)
Schrijf \(113\) met de cijfers van \(0\) tot \(9\)
\(\bbox[3px,border:1px solid blue]{\;Oplossing\;}\)
\(113=25+70+{\Large{36\over4}}+{\Large{81\over9}}\)

113.6
\(113\) als resultaat met breuken waarin de cijfers van \(0\) tot \(9\) exact één keer voorkomen : (\(3\) oplossingen) :
\(409851/3627=532908/4716=809532/7164=113\) 		113.7
Men moet \(113\) tot minimaal de \(43571\)ste macht verheffen opdat in de decimale expansie exact \(113\) \(113\)'s verschijnen.
Terloops : \(113\)\(^{43571}\) is \(89455\) cijfers lang. Noteer dat \(43571\) en \(89455\) exact één maal voorkomen in de decimale expansie. 		113.8
\(113\) is het kleinste priemgetal van drie cijfers met de combinatie “\(11\)”. De andere zijn \(211, 311, 811\) en \(911\).
(OEIS A166572) 		113.9

\(113\) is de noemer van de breuk \(355/113=3,141592{\color{red}{9}}\ldots\) hetgeen een goede benadering voor \(\Large{\pi}\) is.

(de echte waarde van \(\Large{\pi}\) is \(3,141592{\color{blue}{6}}\ldots\))

113.10

De eerste keer dat er \(113\) opeenvolgende samengestelde getallen voorkomen gebeurt tussen de priemgetallen \(492113\)
en \(492227\) met aldus een priemkloof van \(114\,.~~\) (OEIS A000101.pdf)

113.11
\(113\) is het kleinste driecijferpriemgetal wiens cijferproduct en cijfersom allebei priem zijn :
\(1*1*3=3~~\) en \(~~1+1+3=5\) 		113.12

\(b\mathbf{\color{blue}{\;=\;}}113\to b\)\(^{6}\)\(+b\)\(^{0}\)\(+b\)\(^{3}\)\(+b\)\(^{5}\)\(+b\)\(^{1}\)\(+b\)\(^{4}\)\(+b\)\(^{3}\)\(+b\)\(^{0}\)\(+b\)\(^{5}\)\(+b\)\(^{8}\)\(+b\)\(^{4}\)\(+b\)\(^{9}\)\(+b\)\(^{9}\)\(+b\)\(^{2}\)\(+b\)\(^{5}\)\(+b\)\(^{7}\)\(+b\)\(^{2}\)\(+b\)\(^{6}\)\(+b\)\(^{7}\)\(\mathbf{\color{blue}{\;=\;}}6035143058499257267~~\)
(OEIS A236067)

113.13

 ○–○–○ 

\(113^2=12769~~\) en \(~~prime(1+2*7+6+9)=113\)
\(113^3=1442897~~\) en \(~~?=113\)
\(113^4=163047361~~\) en \(~~?=113\)
\(113^5=18424351793~~\) en \(~~?=113\)
\(113^6=2081951752609~~\) en \(~~?=113\)
\(113^7=235260548044817~~\) en \(~~?=113\)
\(113^8=26584441929064321~~\) en \(~~?=113\)
\(113^9=3004041937984268273~~\) en \(~~?=113\)
113.14

Som Der Cijfers (\(sdc\)) van \(k^{\large{113}}\) is gelijk aan het grondtal \(k\). De triviale oplossingen \(0\) en \(1\) negerend vinden we :

\(\qquad\qquad~sdc\left(1548^{\large{113}}\right)=1548\qquad\qquad~sdc\left(1674^{\large{113}}\right)=1674\qquad\qquad~sdc\left(1738^{\large{113}}\right)=1738\)

113.15

Expressie met tweemaal de cijfers uit het getal \(113\) enkel met operatoren \(+,-,*,/,(),\)^^\(\)
\(113\mathbf{\color{blue}{\;=\;}}(1\)^^\(1\)^^\(3)*(3-1-1)\)

113.16

Als expressie met enkelcijferige toepassing, resp. van \(1\) tot \(9~~\) (met dank aan Inder. J. Taneja).
\(\qquad\qquad113=111+1+1\)
\(\qquad\qquad113=2+222/2\)
\(\qquad\qquad113=3+(333-3)/3\)
\(\qquad\qquad113=(444+4+4)/4\)
\(\qquad\qquad113=(555+5+5)/5\)
\(\qquad\qquad113=(666+6+6)/6\)
\(\qquad\qquad113=(7+7+777)/7\)
\(\qquad\qquad113=(888+8+8)/8\)
\(\qquad\qquad113=((9+9)+999)/9\)

113.17

Met de cijfers van \(1\) tot \(9\) in stijgende en dalende volgorde (met dank aan Inder. J. Taneja) :
\(\qquad\qquad113=12+3*4+5+67+8+9\)
\(\qquad\qquad113=9+8+76+5+4*3+2+1\)

113.18

Som der reciproken van partitiegetallen van \(113\) is \(1\) op \(267\) (tweehonderdzevenenzestig) wijzen.

Vijftien partities hebben unieke termen.

\(~~~~(1)~~\bbox[navajowhite,3px,border:1px solid]{113=2+3+8+40+60}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{3}}+{\Large\frac{1}{8}}+{\Large\frac{1}{40}}+{\Large\frac{1}{60}}\)

\(~~~~(3)~~\bbox[navajowhite,3px,border:1px solid]{113=2+4+8+15+24+60}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{4}}+{\Large\frac{1}{8}}+{\Large\frac{1}{15}}+{\Large\frac{1}{24}}+{\Large\frac{1}{60}}\)

\(~~~~(7)~~\bbox[navajowhite,3px,border:1px solid]{113=2+5+8+18+20+24+36}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{5}}+{\Large\frac{1}{8}}+{\Large\frac{1}{18}}+{\Large\frac{1}{20}}+{\Large\frac{1}{24}}+{\Large\frac{1}{36}}\)

\(~~~~(8)~~\bbox[navajowhite,3px,border:1px solid]{113=2+5+9+14+18+30+35}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{5}}+{\Large\frac{1}{9}}+{\Large\frac{1}{14}}+{\Large\frac{1}{18}}+{\Large\frac{1}{30}}+{\Large\frac{1}{35}}\)

\(~~~~(9)~~\bbox[navajowhite,3px,border:1px solid]{113=2+5+9+15+18+24+40}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{5}}+{\Large\frac{1}{9}}+{\Large\frac{1}{15}}+{\Large\frac{1}{18}}+{\Large\frac{1}{24}}+{\Large\frac{1}{40}}\)

\(~~(10)~~\bbox[navajowhite,3px,border:1px solid]{113=2+5+10+11+22+30+33}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{5}}+{\Large\frac{1}{10}}+{\Large\frac{1}{11}}+{\Large\frac{1}{22}}+{\Large\frac{1}{30}}+{\Large\frac{1}{33}}\)

\(~~(11)~~\bbox[navajowhite,3px,border:1px solid]{113=2+5+10+12+18+30+36}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{5}}+{\Large\frac{1}{10}}+{\Large\frac{1}{12}}+{\Large\frac{1}{18}}+{\Large\frac{1}{30}}+{\Large\frac{1}{36}}\)

\(~~(12)~~\bbox[navajowhite,3px,border:1px solid]{113=2+5+10+12+20+24+40}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{5}}+{\Large\frac{1}{10}}+{\Large\frac{1}{12}}+{\Large\frac{1}{20}}+{\Large\frac{1}{24}}+{\Large\frac{1}{40}}\)

\(~~(20)~~\bbox[navajowhite,3px,border:1px solid]{113=2+6+8+12+15+30+40}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{6}}+{\Large\frac{1}{8}}+{\Large\frac{1}{12}}+{\Large\frac{1}{15}}+{\Large\frac{1}{30}}+{\Large\frac{1}{40}}\)

\(~~(52)~~\bbox[navajowhite,3px,border:1px solid]{113=3+4+8+10+15+21+24+28}~~\) en \(~~1={\Large\frac{1}{3}}+{\Large\frac{1}{4}}+{\Large\frac{1}{8}}+{\Large\frac{1}{10}}+{\Large\frac{1}{15}}+{\Large\frac{1}{21}}+{\Large\frac{1}{24}}+{\Large\frac{1}{28}}\)

\(~~(56)~~\bbox[navajowhite,3px,border:1px solid]{113=3+4+9+10+14+18+20+35}~~\) en \(~~1={\Large\frac{1}{3}}+{\Large\frac{1}{4}}+{\Large\frac{1}{9}}+{\Large\frac{1}{10}}+{\Large\frac{1}{14}}+{\Large\frac{1}{18}}+{\Large\frac{1}{20}}+{\Large\frac{1}{35}}\)

\(~~(66)~~\bbox[navajowhite,3px,border:1px solid]{113=3+5+6+10+15+18+20+36}~~\) en \(~~1={\Large\frac{1}{3}}+{\Large\frac{1}{5}}+{\Large\frac{1}{6}}+{\Large\frac{1}{10}}+{\Large\frac{1}{15}}+{\Large\frac{1}{18}}+{\Large\frac{1}{20}}+{\Large\frac{1}{36}}\)

\(~~(72)~~\bbox[navajowhite,3px,border:1px solid]{113=3+5+8+9+10+18+20+40}~~\) en \(~~1={\Large\frac{1}{3}}+{\Large\frac{1}{5}}+{\Large\frac{1}{8}}+{\Large\frac{1}{9}}+{\Large\frac{1}{10}}+{\Large\frac{1}{18}}+{\Large\frac{1}{20}}+{\Large\frac{1}{40}}\)

\((105)~~\bbox[navajowhite,3px,border:1px solid]{113=4+5+6+8+9+12+24+45}~~\) en \(~~1={\Large\frac{1}{4}}+{\Large\frac{1}{5}}+{\Large\frac{1}{6}}+{\Large\frac{1}{8}}+{\Large\frac{1}{9}}+{\Large\frac{1}{12}}+{\Large\frac{1}{24}}+{\Large\frac{1}{45}}\)

\((153)~~\bbox[navajowhite,3px,border:1px solid]{113=4+5+8+9+10+15+18+20+24}~~\) en \(~~1={\Large\frac{1}{4}}+{\Large\frac{1}{5}}+{\Large\frac{1}{8}}+{\Large\frac{1}{9}}+{\Large\frac{1}{10}}+{\Large\frac{1}{15}}+{\Large\frac{1}{18}}+{\Large\frac{1}{20}}+{\Large\frac{1}{24}}\)

(OEIS A125726)

113.19
Het kleinste getal dat exact \(113\) delers heeft is \(5192296858534827628530496329220096=2^{112}\). (OEIS A005179) 113.20
\(113\) is een priemgetal dat het vaakst voorkomt als de \(7\)de priemfactor van een geheel getal. (OEIS A194156) 113.21
Met de cijfers van \(113\) in willekeurige volgorde geschikt bekomt men steeds een priemgetal : \(113,131,311\) zijn
priemgetallen. De drie andere getallen die dezelfde eigenschap hebben, zijn en .
Lees er meer over bij Circular Primes 		113.22

\(113^7\mathbf{\color{blue}{\;=\;}}9262^3+15312283^2\) is de enige oplossing met positieve getallen voor \(a^7=b^3+c^2\)

113.23

(twee multigrades) \(113\to113^5\to\)

\begin{aligned} 113^1&=61^1-74^1+129^1+179^1-182^1\\ 113^5&=61^5-74^5+129^5+179^5-182^5\\ \\ 113^1&=-124^1-373^1+619^1+1163^1-1172^1\\ 113^5&=-124^5-373^5+619^5+1163^5-1172^5\\ \end{aligned}

113.24

Er zijn \(113\) palindroom priemgetallen kleiner dan \(10^6\).

\(\qquad93\) getallen van \(5\) cijfers

\(\qquad0\) getallen van \(4\) cijfers

\(\qquad15\) getallen van \(3\) cijfers

\(\qquad1\) getal van \(2\) cijfers

\(\qquad4\) getallen van \(1\) cijfer

113.25

Kleinste oplossing voor de positieve Pell vergelijking \(x^2-D*y^2\mathbf{\color{blue}{\;=\;}}1~\) met \(D\mathbf{\color{blue}{\;=\;}}113\).

Als \(D\) een kwadraat is dan zijn er geen oplossingen.

\(\qquad{\color{darkviolet}{1204353}}^2-113*{\color{darkviolet}{113296}}^2\mathbf{\color{blue}{\;=\;}}1\)

(Pell equation solver)

113.26
De reciprook van \(113\) heeft als decimale periode de maximale waarde \(DP(1/113)\mathbf{\color{blue}{\;=\;}}113-1\mathbf{\color{blue}{\;=\;}}112\).
De volledige decimale expansie van \(1\) periode na de komma is
\(00884955752212389380530973451327433628318584070796460176{\color{darkcyan}{9911504424778761061946902654867256}}\)
\({\color{darkcyan}{6371681415929203539823}}\)

(OEIS A001913)

Splitst men deze periode van \(112\) cijfers in twee gelijke groepen van \(56\) cijfers dan is de som van elk tweetal cijfers
onder elkaar steeds \(9\).

113.27
Schakelaar
\(\mathbf[0\gets\to1000\mathbf]\)
“Allemaal Getallen”

⏮

⯬

⏴

⏵

⯮

⏭

\(113\)\(_{\large\color{green}{30}}\)\(113\)\(2\)\(114\)
\(1,113\)
Priemgetal\(1110001_2\)\(71_{16}\)
   

Uit de collectie 'Allemaal Getallen' van Ir. Jos Heynderickx
Toevoegingen & Bewerking & Layout door Patrick De Geest (email)
Laatste update 6 maart 2026