\(140\mathbf{\color{blue}{\;=\;}}14+15+16+17+18+19+20+21\mathbf{\color{blue}{\;=\;}}17+18+19+20+21+22+23\mathbf{\color{blue}{\;=\;}}26+27+28+29+30\)

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

\(140\mathbf{\color{blue}{\;=\;}}14+16+18+20+22+24+26\mathbf{\color{blue}{\;=\;}}24+26+28+30+32\mathbf{\color{blue}{\;=\;}}32+34+36+38\)

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

\(140\mathbf{\color{blue}{\;=\;}}5+7+9+11+13+15+17+19+21+23\mathbf{\color{blue}{\;=\;}}69+71\) (som van opeenvolgende onpare getallen)

\(140\mathbf{\color{blue}{\;=\;}}5*(1+2+3+4+5+6+7)\) (vijf maal de som van opeenvolgende gehele getallen)

\(140\mathbf{\color{blue}{\;=\;}}1^2+2^2+3^2+4^2+5^2+6^2+7^2\mathbf{\color{blue}{\;=\;}}1+4+9+16+25+36+49\)

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

\(140\mathbf{\color{blue}{\;=\;}}((0;2;6;10)\,(1;3;3;11)\,(1;3;7;9)\,(2;6;6;8)\,(3;5;5;9))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#5\}\)

\(140\mathbf{\color{blue}{\;=\;}}((0;0;1;1;1;1;2;4;4)\,(0;2;2;2;2;3;3;3;3)\,(1;1;1;1;1;1;1;2;5)\,(1;2;2;2;2;2;2;3;4))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#4\}\)

\(140\mathbf{\color{blue}{\;=\;}}(1+4+0+1^3+4^3+0^3)*2\)

\(140\mathbf{\color{blue}{\;=\;}}eulerphi(213,284,426)\)

\(140\mathbf{\color{blue}{\;=\;}}tribonacci_{(0,1,5)}[9]\mathbf{\color{blue}{\;=\;}}23+41+76\)

\(140\mathbf{\color{blue}{\;=\;}}tetranacci_{(0,1,7,2)}[9]\mathbf{\color{blue}{\;=\;}}10+20+39+71\)

\(140\mathbf{\color{blue}{\;=\;}}[6^4][36^2]-34^2\mathbf{\color{blue}{\;=\;}}12^2-2^2\)

\(140\mathbf{\color{blue}{\;=\;}}\)(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 \(m\mathbf{\color{blue}{\;=\;}}15~~(+5)\).

\(140\mathbf{\color{blue}{\;=\;}}\)(som van vier derdemachten)

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

\(140\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

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

\(\qquad~~~~\bbox[3px,border:1px blue solid]{(-1)^5+(-3)^5+26^5+32^5+(-34)^5}\mathbf{\color{blue}{\;=\;}}\)

\(140^2\mathbf{\color{blue}{\;=\;}}24^3+76^2\mathbf{\color{blue}{\;=\;}}65^3-505^2\mathbf{\color{blue}{\;=\;}}84^2+112^2\mathbf{\color{blue}{\;=\;}}148^2-48^2\mathbf{\color{blue}{\;=\;}}149^2-51^2\mathbf{\color{blue}{\;=\;}}175^2-105^2\mathbf{\color{blue}{\;=\;}}203^2-147^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;221^2-171^2\mathbf{\color{blue}{\;=\;}}265^2-[15^4][225^2]\mathbf{\color{blue}{\;=\;}}364^2-336^2\mathbf{\color{blue}{\;=\;}}500^2-480^2\mathbf{\color{blue}{\;=\;}}707^2-693^2\mathbf{\color{blue}{\;=\;}}985^2-975^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;1085^2-105^3\mathbf{\color{blue}{\;=\;}}1229^2-1221^2\mathbf{\color{blue}{\;=\;}}2452^2-2448^2\mathbf{\color{blue}{\;=\;}}4901^2-4899^2\)

\(140^3\mathbf{\color{blue}{\;=\;}}119^3+1029^2\mathbf{\color{blue}{\;=\;}}329^3-5733^2\mathbf{\color{blue}{\;=\;}}1659^2-91^2\mathbf{\color{blue}{\;=\;}}1680^2-280^2\mathbf{\color{blue}{\;=\;}}1686^2-314^2\mathbf{\color{blue}{\;=\;}}1785^2-665^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;1872^2-872^2\mathbf{\color{blue}{\;=\;}}1890^2-910^2\mathbf{\color{blue}{\;=\;}}2115^2-1315^2\mathbf{\color{blue}{\;=\;}}2142^2-1358^2\mathbf{\color{blue}{\;=\;}}2310^2-1610^2\mathbf{\color{blue}{\;=\;}}2343^2-1657^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;2730^2-2170^2\mathbf{\color{blue}{\;=\;}}2994^2-2494^2\mathbf{\color{blue}{\;=\;}}3045^2-2555^2\mathbf{\color{blue}{\;=\;}}3630^2-3230^2\mathbf{\color{blue}{\;=\;}}3696^2-3304^2\mathbf{\color{blue}{\;=\;}}4095^2-3745^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;5040^2-4760^2\mathbf{\color{blue}{\;=\;}}5613^2-5363^2\mathbf{\color{blue}{\;=\;}}6237^2-6013^2\mathbf{\color{blue}{\;=\;}}6960^2-6760^2\mathbf{\color{blue}{\;=\;}}7098^2-6902^2\mathbf{\color{blue}{\;=\;}}8655^2-8495^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;\bbox[2px,border:1px brown dashed]{9870^2-9730^2}\)

\(140\mathbf{\color{blue}{\;=\;}}7!/(3!*3!)\mathbf{\color{blue}{\;=\;}}7!/(6^2)\mathbf{\color{blue}{\;=\;}}5040/36\)

\(140\) is gelijk aan \(28\) maal de som van zijn cijfers : \(140\mathbf{\color{blue}{\;=\;}}28*(1+4+0)\)
Andere getallen met dezelfde eigenschap zijn \(112,224,252,280,308,336,364,392,448,476\) en \(588\).
(OEIS A005349 - Harshad getallen)

Als som met de vier operatoren \(+-*\;/\)
\(140\mathbf{\color{blue}{\;=\;}}(35+1)+(35-1)+(35*1)+(35/1)\)

\(\begin{align}140\mathbf{\color{blue}{\;=\;}}\left({\frac{6623}{5301}}\right)^3+\left({\frac{27397}{5301}}\right)^3\end{align}\)

(Integral Sum of Two Rational Cubes) (OEIS A020898) (OEIS A228499) (Links uit OEIS A060838)

\((x^3+y^3)/z^3\mathbf{\color{blue}{\;=\;}}n~\to~\) [x waarde] (OEIS A190356)  [y waarde] (OEIS A190580)  [z waarde] (OEIS A190581)

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

Som Der Cijfers (\(sdc\)) van \(k^{\large{140}}\) is gelijk aan het grondtal \(k\). De triviale oplossingen \(0\) en \(1\) negerend vinden we :

\(\qquad\qquad~sdc\left(1270^{\large{140}}\right)\mathbf{\color{blue}{\;=\;}}1270\qquad\qquad~sdc\left(2116^{\large{140}}\right)\mathbf{\color{blue}{\;=\;}}2116\qquad\qquad~sdc\left(2178^{\large{140}}\right)\mathbf{\color{blue}{\;=\;}}2178\)

\(\qquad\qquad~sdc\left(2214^{\large{140}}\right)\mathbf{\color{blue}{\;=\;}}2214\)

Expressie met tweemaal de cijfers uit het getal \(140\) bij voorkeur met de basisoperatoren
\(140\mathbf{\color{blue}{\;=\;}}(1\)^^\(4)*(1\)^^\(0)+4*0\mathbf{\color{blue}{\;=\;}}(1\)^^\(4\)^^\(0)+1*4*0\)

Als expressie met enkelcijferige toepassing, resp. van \(1\) tot \(9~~\) (met dank aan Inder. J. Taneja).
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}(11-1)*(11+1+1+1)\)
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}2*(2*(22+2)+22)\)
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}3+3^3+(333-3)/3\)
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}4*(4*(4+4)+4)-4\)
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}5*5*5+5+5+5\)
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}66+66+6+(6+6)/6\)
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}7*(7+7+7)-7\)
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}8*(8+8)+(88+8)/8\)
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}9+9+(999+99)/9\)

Met de cijfers van \(1\) tot \(9\) in stijgende en dalende volgorde (met dank aan Inder. J. Taneja) :
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}12+3+4+56+7*8+9\)
\(\qquad\qquad140\mathbf{\color{blue}{\;=\;}}9*8+7*6+5*4+3+2+1\)

\(140\) is een harmonisch-delergetal.

De teller bevat het aantal delers van \(140\) ofwel \({\large\sigma}(140)\mathbf{\color{blue}{\;=\;}}12\).

De noemer is de harmonische reeks van de \(12\) delers i.e. de som van de reciproken van alle delers.

Als de teller gedeeld door de noemer een geheel getal is dan zeggen we dat \(140\) een harmonisch-delergetal is.

\({\Large\frac{12}{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\frac{1}{7}+\frac{1}{10}+\frac{1}{14}+\frac{1}{20}+\frac{1}{28}+\frac{1}{35}+\frac{1}{70}+\frac{1}{140}}}\mathbf{\color{blue}{\;=\;}}5\)

(Harmonisch-delergetal) (Harmonic divisor number) (OEIS A001599)

Zes kwadraten die dezelfde cijfers gebruiken in hun decimale expansie :

\begin{align} 103^2&\mathbf{\color{blue}{\;=\;}}10609\\ 130^2&\mathbf{\color{blue}{\;=\;}}16900\\ {\color{blue}{140}}^2&\mathbf{\color{blue}{\;=\;}}19600\\ 247^2&\mathbf{\color{blue}{\;=\;}}61009\\ 301^2&\mathbf{\color{blue}{\;=\;}}90601\\ 310^2&\mathbf{\color{blue}{\;=\;}}96100 \end{align}

(vier multigrades) \(140\to140^5\to\)

\begin{aligned} 140^1&\mathbf{\color{blue}{\;=\;}}-42^1+65^1+94^1-129^1+152^1\\ 140^5&\mathbf{\color{blue}{\;=\;}}-42^5+65^5+94^5-129^5+152^5\\ \\ 140^1&\mathbf{\color{blue}{\;=\;}}-96^1-160^1+436^1+508^1-548^1\\ 140^5&\mathbf{\color{blue}{\;=\;}}-96^5-160^5+436^5+508^5-548^5\\ \\ 140^1&\mathbf{\color{blue}{\;=\;}}195^1+710^1-835^1-860^1+930^1\\ 140^5&\mathbf{\color{blue}{\;=\;}}195^5+710^5-835^5-860^5+930^5\\ \\ 140^1&\mathbf{\color{blue}{\;=\;}}-80^1-845^1+1165^1+1310^1-1410^1\\ 140^5&\mathbf{\color{blue}{\;=\;}}-80^5-845^5+1165^5+1310^5-1410^5\\ \end{aligned}

Kleinste oplossing voor de positieve Pell vergelijking \(x^2-D*y^2\mathbf{\color{blue}{\;=\;}}1~\) met \(D\mathbf{\color{blue}{\;=\;}}140\).

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

\(\qquad{\color{darkviolet}{71}}^2-140*{\color{darkviolet}{6}}^2\mathbf{\color{blue}{\;=\;}}1\)

(Pell equation solver)

(OEIS A001913) (OEIS A051626) (OEIS A060284)

("In case anyone was curious - inverse.txt" by Philip Clarke)

Splitst men deze periode van \(6\) cijfers in twee gelijke groepen van \(3\) cijfers dan is de som van elk tweetal cijfers
onder elkaar steeds \(9\).

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