\(113=56+57\) (som van opeenvolgende gehele getallen)

\(113=49+64\) (som van opeenvolgende kwadraatgetallen)

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

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

\(113=2^4+2^4+3^4\)

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

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

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)

\(\qquad~~~~\bbox[lightyellow,3px,border:1px blue solid]{~oplossing~onbekend~}\mathbf{\color{blue}{\;=\;}}(z\gt200)\)

113.2

\(113\) 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 \(3,141592{\color{blue}{6}}\ldots\))

113.3

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

\(113^2=12769\) en het omgekeerde \(96721=311^2\). Hetzelfde doet zich voor bij

\(113^7=9262^3+15312283^2\) is de enige oplossing met positieve getallen voor \(a^7=b^3+c^2\)

113.4
Met de cijfers van \(113\) in willekeurige volgorde geschikt bekomt men steeds een priemgetal : \(113,131,311\) zijn priemgetallen. De twee andere getallen die dezelfde eigenschap hebben, zijn \(199\) en \(337\).
Lees er meer over bij Circular Primes
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

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

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

113.10

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 driecijferig priemgetal 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
Schakelaar
\(\mathbf[0\gets\to1000\mathbf]\)
Allemaal Getallen

\(\Huge\bbox[border:0]{⏮}\)

\(\Huge\bbox[border:0]{⯬}\)

\(\Huge\bbox[border:0]{⏴}\)

\(\Huge\bbox[border:0]{⏵}\)

\(\Huge\bbox[border:0]{⯮}\)

\(\Huge\bbox[border:0]{⏭}\)


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

Uit de collectie 'Allemaal Getallen' van Ir. Jos Heynderickx
Toevoegingen & Bewerking & Layout door Patrick De Geest (email)
Laatste update 7 december 2024