\(136=1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16\)

\(\qquad~~~~\)(som van opeenvolgende gehele getallen)

\(136=10+12+14+16+18+20+22+24\) (som van opeenvolgende pare getallen)

\(136\mathbf{\color{blue}{\;=\;}}31+33+35+37\mathbf{\color{blue}{\;=\;}}67+69\) (som van opeenvolgende onpare getallen)

\(136=5+8+13+21+34+55\) (som van opeenvolgende Fibonaccigetallen)

\(136\mathbf{\color{blue}{\;=\;}}36+45+55\mathbf{\color{blue}{\;=\;}}D(8)+D(9)+D(10)\) (som van opeenvolgende driehoeksgetallen)

\(136=((0;0;6;10)\,(0;6;6;8)\,(2;2;8;8)\,(2;4;4;10))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#4\}\)

\(136\mathbf{\color{blue}{\;=\;}}\bbox[#f4f4f4,3px,border:1px solid]{2^3+4^3+4^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~((0;0;0;0;0;0;2;4;4)\,(0;0;0;0;1;1;1;2;5)\,(0;0;0;1;3;3;3;3;3)\,(0;0;1;1;2;2;3;3;4))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#4\}\)

\(136\mathbf{\color{blue}{\;=\;}}(1^2+1^2)*(2^2+8^2)\mathbf{\color{blue}{\;=\;}}(1^2+4^2)*(2^2+2^2)\)

\(136=(1+3+6)*13+6\)

\(136=17*(1+7)\)

\(136\mathbf{\color{blue}{\;=\;}}2^3+2^7\mathbf{\color{blue}{\;=\;}}6^2+10^2\mathbf{\color{blue}{\;=\;}}12^2-2^3\mathbf{\color{blue}{\;=\;}}19^2-15^2\mathbf{\color{blue}{\;=\;}}35^2-33^2\)

\(136\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~17\) oplossingen bekend

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

\(136\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

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

\(136\to4646\to\) \begin{align} 30^1+35^1+35^1+36^1&=32^1+33^1+33^1+38^1\\ &en\\ 30^2+35^2+35^2+36^2&=32^2+33^2+33^2+38^2 \end{align}

\(136=2^3+4^3+4^3~~\) en \(~~244=1^3+3^3+6^3\). Hetzelfde patroon vindt men met het getal 919.-- (zie aldaar).

\(136=2^1+3^2+5^3\)

\(136^2\mathbf{\color{blue}{\;=\;}}[2^{12}][4^6][8^4][16^3][64^2]+120^2\mathbf{\color{blue}{\;=\;}}17^4-255^2\mathbf{\color{blue}{\;=\;}}68^3-544^2\mathbf{\color{blue}{\;=\;}}153^2-17^3\mathbf{\color{blue}{\;=\;}}170^2-102^2\mathbf{\color{blue}{\;=\;}}289^2-255^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;305^2-273^2\mathbf{\color{blue}{\;=\;}}586^2-570^2\mathbf{\color{blue}{\;=\;}}1160^2-1152^2\mathbf{\color{blue}{\;=\;}}2314^2-2310^2\mathbf{\color{blue}{\;=\;}}4488^2-272^3\mathbf{\color{blue}{\;=\;}}4625^2-4623^2\)

\(136^3\mathbf{\color{blue}{\;=\;}}71^4-4785^2\mathbf{\color{blue}{\;=\;}}80^2+1584^2\mathbf{\color{blue}{\;=\;}}816^2+1360^2\mathbf{\color{blue}{\;=\;}}1666^2-510^2\mathbf{\color{blue}{\;=\;}}1700^2-612^2\mathbf{\color{blue}{\;=\;}}2465^2-1887^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;2584^2-2040^2\mathbf{\color{blue}{\;=\;}}4760^2-4488^2\mathbf{\color{blue}{\;=\;}}5041^2-4785^2\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{9316^2-9180^2}\mathbf{\color{blue}{\;=\;}}9890^2-9762^2\)

Als som met de vier operatoren \(+-*\;/\)
\(136=(34+1)+(34-1)+(34*1)+(34/1)\)

\({\color{blue}{136^2}}+137^2+138^2+139^2+140^2+141^2+142^2+143^2+144^2=\)

\(145^2+146^2+147^2+148^2+149^2+150^2+151^2+152^2={\color{tomato}{176460}}\)

Gelijkheid van twee sommen met doorlopend opeenvolgende kwadraten.

De eerste term van de linkersom is (OEIS A014105).

De eerste term \(m\) van de rechtersom is (OEIS A001804). \(2*m-1=2*145-1=289=17^2\) is een perfect kwadraat.

Het verschil \(145-136=9\) geeft het aantal termen weer van de linkersom. En ééntje minder in de rechtersom.

(OEIS A059255)

\(\begin{align}136\mathbf{\color{blue}{\;=\;}}\left({\frac{36}{7}}\right)^3-\left({\frac{2}{7}}\right)^3\mathbf{\color{blue}{\;=\;}}\color{tomato}{\left({\frac{18*2}{7}}\right)^3-\left({\frac{1*2}{7}}\right)^3\mathbf{\color{blue}{\;=\;}}\left({\frac{18}{7}}\right)^3-\left({\frac{1}{7}}\right)^3\mathbf{\color{blue}{\;=\;}}{\frac{136}{8}}\mathbf{\color{blue}{\;=\;}}17}\end{align}\)

(Integral Sum of Two Rational Cubes) (OEIS A020898) (OEIS A228499)

\((x^3+y^3)/z^3=n~\to~\) [x waarde] (OEIS A190356)  [y waarde] (OEIS A190580)  [z waarde] (OEIS A190581)

Kleinste positieve oplossingen \(~\to~\) [x waarde] (OEIS A254326)  [y waarde] (OEIS A254324)

\(989353^3=968397868897889977~~\) en

\(9+6+8+3+9+7+8+6+8+8+9+7+8+8+9+9+7+7={\color{blue}{136}}\) en

\(989353\) is het hoogste getal kleiner dan een miljoen met die eigenschap.