\(289\mathbf{\color{blue}{\;=\;}}9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25\mathbf{\color{blue}{\;=\;}}144+145\)

\(\qquad~~~~\)(som van opeenvolgende gehele getallen op twee wijzen)

\(289\mathbf{\color{blue}{\;=\;}}1+3+5+7+9+11+13+15+17+19+21+23+25+27+29+31+33\)

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

\(289\mathbf{\color{blue}{\;=\;}}136+153\mathbf{\color{blue}{\;=\;}}D(16)+D(17)\) (som van opeenvolgende driehoeksgetallen)

\(289\mathbf{\color{blue}{\;=\;}}2^1+2^2+3^3+4^4\mathbf{\color{blue}{\;=\;}}0^0+1^1+2^2+3^3+4^4\)

\(289\mathbf{\color{blue}{\;=\;}}((0;0;0;17)\,(0;0;8;15)\,(0;1;12;12)\,(0;8;9;12)\,(1;4;4;16)\,(2;2;5;16)\,(2;4;10;13)\)

\(\qquad~~~~(2;5;8;14)\,(2;8;10;11)\,(3;6;10;12)\,(5;8;10;10))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#11\}\)

\(289\mathbf{\color{blue}{\;=\;}}((0;0;0;0;0;1;2;4;6)\,(0;0;0;1;2;3;4;4;5)\,(0;1;1;1;1;2;3;5;5)\,(0;1;1;1;2;2;3;3;6)\)

\(\qquad~~~~(0;2;2;3;3;3;4;4;4)\,(1;1;3;3;3;3;3;3;5)\,(1;2;2;2;2;4;4;4;4))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#7\}\)

\(289\mathbf{\color{blue}{\;=\;}}2^3+2^8+5^2\)

\(289\mathbf{\color{blue}{\;=\;}}5^3+8^2+10^2\)

\(289\mathbf{\color{blue}{\;=\;}}(2+3+5+7){\large^2}~~\) (kwadraat van de som van de vier eerste priemgetallen)

\(289\mathbf{\color{blue}{\;=\;}}28*9+28+9\)

\(289\mathbf{\color{blue}{\;=\;}}(8+9)^2\)

\(289\mathbf{\color{blue}{\;=\;}}tribonacci_{(1,3,9)}[9]\mathbf{\color{blue}{\;=\;}}47+85+157\)

\(289\mathbf{\color{blue}{\;=\;}}tetranacci_{(0,2,9,0)}[10]\mathbf{\color{blue}{\;=\;}}22+42+75+150\)

\(289\mathbf{\color{blue}{\;=\;}}[2^6][4^3][8^2]+15^2\mathbf{\color{blue}{\;=\;}}17^3-68^2\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{145^2-[12^4][144^2]}\mathbf{\color{blue}{\;=\;}}561^2-68^3\)

\(289\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~10\) oplossingen bekend

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

\(289\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

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

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

\(289^2\mathbf{\color{blue}{\;=\;}}17^5-[34^4][1156^2]\mathbf{\color{blue}{\;=\;}}136^2+255^2\mathbf{\color{blue}{\;=\;}}161^2+240^2\mathbf{\color{blue}{\;=\;}}397^2-42^3\mathbf{\color{blue}{\;=\;}}2465^2-2448^2\)

\(289^3\mathbf{\color{blue}{\;=\;}}495^2+4888^2\mathbf{\color{blue}{\;=\;}}2312^2+4335^2\mathbf{\color{blue}{\;=\;}}2737^2+4080^2\mathbf{\color{blue}{\;=\;}}\cdots\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{41905^2-41616^2}\)

De eerste keer dat er \(289\) opeenvolgende samengestelde getallen voorkomen gebeurt tussen de priemgetallen
\(1948819133~\) en \(~1948819423\) met aldus een priemkloof van \(290\,.~~\) (OEIS A000101.pdf)

\(\begin{align}289\mathbf{\color{blue}{\;=\;}}\left({\frac{199}{31}}\right)^3+\left({\frac{90}{31}}\right)^3\end{align}\)

(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)

\({\color{blue}{289}}+290+291+292+293+294+295+296+297+298+299+300+301+302+303+304+305+306\mathbf{\color{blue}{\;=\;}}\)

\(307+308+309+310+311+312+313+314+315+316+317+318+319+320+321+322+323\mathbf{\color{blue}{\;=\;}}\)\(~~{\color{tomato}{5355}}\)

Gelijkheid van twee sommen met doorlopend opeenvolgende gehele getallen.

Dit is een eigenschap van de kwadraten zoals \(k\mathbf{\color{blue}{\;=\;}}289\mathbf{\color{blue}{\;=\;}}17^2\). De linkersom heeft \(\sqrt{k}+1\) termen en de rechtersom \(\sqrt{k}\).

(OEIS A059270)

Product van \(153\), een driehoeksgetal van rang \(17\), en de som van de reciproken van de \(17\) kleinste driehoeksgetallen :

\(153*\left({\Large{\frac{1}{1}}+{\frac{1}{3}}+{\frac{1}{6}}+{\frac{1}{10}}+\cdots+{\frac{1}{136}}+{\frac{1}{153}}}\right)\mathbf{\color{blue}{\;=\;}}289\).

Pari/GP code : 153*sum(d=1,17,1/((d*d+d)/2))

\(289\mathbf{\color{blue}{\;=\;}}17^2\mathbf{\color{blue}{\;=\;}}{\Large\frac{18!\,-\,17!}{16!}}\)

 ○–○–○ 

Som Der Cijfers (\(sdc\)) van \(k^{\large{289}}\) is gelijk aan het grondtal \(k\). De triviale oplossingen \(0\) en \(1\) negerend vinden we :

\(\qquad\qquad~sdc\left(4498^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4498\qquad\qquad~sdc\left(4676^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4676\qquad\qquad~sdc\left(4744^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4744\)

\(\qquad\qquad~sdc\left(4786^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4786\qquad\qquad~sdc\left(4787^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4787\qquad\qquad~sdc\left(4795^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4795\)

\(\qquad\qquad~sdc\left(4813^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4813\qquad\qquad~sdc\left(4833^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4833\qquad\qquad~sdc\left(4844^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4844\)

\(\qquad\qquad~sdc\left(4847^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4847\qquad\qquad~sdc\left(4891^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}4891\qquad\qquad~sdc\left(5027^{\large{289}}\right)\mathbf{\color{blue}{\;=\;}}5027\)

Expressie met tweemaal de cijfers uit het getal \(289\) bij voorkeur met de basisoperatoren
\(289\mathbf{\color{blue}{\;=\;}}((8*8)+(2/9))*(9/2)\mathbf{\color{blue}{\;=\;}}(2\)^^\(8)*9+(2\)^^\(8)+9\mathbf{\color{blue}{\;=\;}}(2\)^^\(8\)^^\(9)*(2+8-9)\)

Als expressie met enkelcijferige toepassing, resp. van \(1\) tot \(9~~\) (met dank aan Inder. J. Taneja).
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}(1+1)*(11+1)^{(1+1)}+1\)
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}(2^{(2+2)}+2/2)^2\)
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}3*(3*33-3)+3/3\)
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}44+4^4-44/4\)
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}5*(55+5)-55/5\)
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}6*(6*6+6+6)+6/6\)
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}7*(7*7-7)-7+(7+7)/7\)
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}(8+8+8/8)^{((8+8)/8)}\)
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}(9+9-9/9)^{((9+9)/9)}\)

Met de cijfers van \(1\) tot \(9\) in stijgende en dalende volgorde (met dank aan Inder. J. Taneja) :
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}1+234+5*6+7+8+9\)
\(\qquad\qquad289\mathbf{\color{blue}{\;=\;}}9+8+7+65*4+3+2*1~~{\small\color{green}{(Origineel~{\color{red}{98+7+65*4+3+21}}~is~oplossing~voor~389)}}\)

(multigrades) \(289\to289^5\to\)

\begin{aligned} 289^1&\mathbf{\color{blue}{\;=\;}}7^1-46^1+350^1+403^1-425^1\\ 289^5&\mathbf{\color{blue}{\;=\;}}7^5-46^5+350^5+403^5-425^5\\ \end{aligned}

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

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

Splitst men deze periode van \(272\) cijfers in twee gelijke groepen van \(136\) cijfers dan is de som van elk tweetal cijfers
onder elkaar steeds \(9\).

\({\Large\left(\frac{289}{2}\right)^3}\mathbf{\color{blue}{\;=\;}}{\Large\frac{24137569}{8}}~~\) en dat is een pandigitale breuk met cijfers van \(1\) tot \(9\)

\(289\mathbf{\color{blue}{\;=\;}}17^2\) kan uitgedrukt worden als de som van \(1,2,3,4,5,6,7,8\) verschillende kwadraten.

\begin{align} 17^2&\mathbf{\color{blue}{\;=\;}}17^2\\ 17^2&\mathbf{\color{blue}{\;=\;}}8^2+15^2\\ 17^2&\mathbf{\color{blue}{\;=\;}}8^2+9^2+12^2\\ 17^2&\mathbf{\color{blue}{\;=\;}}2^2+4^2+10^2+13^2\\ 17^2&\mathbf{\color{blue}{\;=\;}}2^2+3^2+4^2+8^2+14^2\\ 17^2&\mathbf{\color{blue}{\;=\;}}1^2+2^2+3^2+5^2+9^2+13^2\\ 17^2&\mathbf{\color{blue}{\;=\;}}1^2+3^2+4^2+5^2+6^2+9^2+11^2\\ 17^2&\mathbf{\color{blue}{\;=\;}}1^2+2^2+3^2+4^2+5^2+7^2+8^2+11^2\\ \end{align}

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