\(267\mathbf{\color{blue}{\;=\;}}\)als som van opeenvolgende gehele getallen op drie verschillende wijzen :

\begin{cases} 267\mathbf{\color{blue}{\;=\;}}42+43+44+45+46+47\\ 267\mathbf{\color{blue}{\;=\;}}88+89+90\\ 267\mathbf{\color{blue}{\;=\;}}65+66+67+68 \end{cases}

\(267\mathbf{\color{blue}{\;=\;}}87+89+91\) (som van opeenvolgende onpare getallen)

\(267\mathbf{\color{blue}{\;=\;}}((0;5;11;11)\,(0;7;7;13)\,(1;1;3;16)\,(1;1;11;12)\,(1;4;5;15)\,(1;4;9;13)\,(1;8;9;11)\)

\(\qquad~~~~(3;4;11;11)\,(3;5;8;13)\,(4;7;9;11)\,(5;7;7;12))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#11\}\)

\(267\mathbf{\color{blue}{\;=\;}}((0;0;0;0;1;2;2;5;5)\,(0;0;0;0;2;2;2;3;6)\,(0;0;2;2;2;3;3;4;5)\,(0;1;1;1;2;4;4;4;4))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#4\}\)

\(267\mathbf{\color{blue}{\;=\;}}2^1+6^3+7^2\)

\(267\mathbf{\color{blue}{\;=\;}}7^2+7^2+(6+7)^2\)

\(267\mathbf{\color{blue}{\;=\;}}fibonacci_{(0,3)}[12]\mathbf{\color{blue}{\;=\;}}102+165\)

\(267\mathbf{\color{blue}{\;=\;}}tribonacci_{(1,3,3)}[10]\mathbf{\color{blue}{\;=\;}}43+79+145\)

\(267\mathbf{\color{blue}{\;=\;}}tetranacci_{(0,3,9,7)}[9]\mathbf{\color{blue}{\;=\;}}19+38+73+137\)

\(267\mathbf{\color{blue}{\;=\;}}46^2-43^2\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{134^2-133^2}\)

\(267\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~7\) oplossingen bekend

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

\(\qquad~~~~\bbox[3px,border:1px solid]{(-4)^3+(-10)^3+11^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px solid]{(-6754)^3+(-8242)^3+9539^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px solid]{14099^3+28757^3+(-29845)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px solid]{11953754^3+(-16659322)^3+14285651^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px solid]{(-3279826657)^3+(-6053866741)^3+6359113961^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px solid]{4527009152^3+9189527498^3+(-9542039077)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px solid]{(-23297228461)^3+(-56896819084)^3+58170124628^3}\mathbf{\color{blue}{\;=\;}}\)

\(267\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

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

\(\qquad~~~~\bbox[3px,border:1px blue solid]{(-4)^5+(-7)^5+9^5+9^5+(-10)^5}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px blue solid]{(-106)^5+(-133)^5+(-162)^5+(-250)^5+258^5}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px blue solid]{350^5+848^5+(-965)^5+(-1216)^5+1250^5}\)

\(267^2\mathbf{\color{blue}{\;=\;}}117^2+240^2\mathbf{\color{blue}{\;=\;}}445^2-356^2\mathbf{\color{blue}{\;=\;}}3965^2-3956^2\)

\(267^3\mathbf{\color{blue}{\;=\;}}5162^2-2759^2\mathbf{\color{blue}{\;=\;}}\cdots\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{35778^2-35511^2}\)

De eerste keer dat er \(267\) opeenvolgende samengestelde getallen voorkomen gebeurt tussen de priemgetallen
\(1579306789~\) en \(~1579307057\) met aldus een priemkloof van \(268\,.~~\) (OEIS A000101.pdf)

\(\begin{align}267\mathbf{\color{blue}{\;=\;}}\left({\frac{861409}{130914}}\right)^3-\left({\frac{342361}{130914}}\right)^3\end{align}\)

(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 \(k^{\large{267}}\) is gelijk aan het grondtal \(k\). De triviale oplossingen \(0\) en \(1\) negerend vinden we :

\(\qquad\qquad~sdc\left(3060^{\large{267}}\right)\mathbf{\color{blue}{\;=\;}}3060\qquad\qquad~sdc\left(4492^{\large{267}}\right)\mathbf{\color{blue}{\;=\;}}4492\qquad\qquad~sdc\left(4537^{\large{267}}\right)\mathbf{\color{blue}{\;=\;}}4537\)

\(\qquad\qquad~sdc\left(4544^{\large{267}}\right)\mathbf{\color{blue}{\;=\;}}4544\)

Expressie met tweemaal de cijfers uit het getal \(267\) enkel met operatoren \(+,-,*,/,(),\)^^\(\)
\(267\mathbf{\color{blue}{\;=\;}}(2\)^^\(6\)^^\(7)*(2+6-7)\)

Als expressie met enkelcijferige toepassing, resp. van \(1\) tot \(9~~\) (met dank aan Inder. J. Taneja).
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}(1+1)*(11*(11+1)+1)+1\)
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}2^{(2*(2+2))}+22/2\)
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}3*3*(3^3+3)-3\)
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}4^4+44/4\)
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}5*55-5-5+(5+5)/5\)
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}6*666/(6+6)-66\)
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}7*7*7-77+7/7\)
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}(8+8)*(8+8)+88/8\)
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}((9+9)/9)^{(9-9/9)}+99/9\)

Met de cijfers van \(1\) tot \(9\) in stijgende en dalende volgorde (met dank aan Inder. J. Taneja) :
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}123+4*5*6+7+8+9\)
\(\qquad\qquad267\mathbf{\color{blue}{\;=\;}}9+87+6+54*3+2+1\)

(multigrades) \(267\to8395412607~~\text{(pannumerisch)}\to\)

\begin{aligned} 19^1+31^1+52^1+77^1+88^1&\mathbf{\color{blue}{\;=\;}}25^1+26^1+50^1+79^1+87^1\\ 19^5+31^5+52^5+77^5+88^5&\mathbf{\color{blue}{\;=\;}}25^5+26^5+50^5+79^5+87^5 \end{aligned}

(multigrades) \(267\to8932546107~~\text{(pannumerisch)}\to\)

\begin{aligned} 4^1+43^1+61^1+67^1+92^1&\mathbf{\color{blue}{\;=\;}}12^1+21^1+75^1+75^1+84^1\\ 4^5+43^5+61^5+67^5+92^5&\mathbf{\color{blue}{\;=\;}}12^5+21^5+75^5+75^5+84^5 \end{aligned}

Kleinste oplossing voor de positieve Pell vergelijking \(x^2-D*y^2\mathbf{\color{blue}{\;=\;}}1~\) met \(D\mathbf{\color{blue}{\;=\;}}267\).

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

\({\qquad\color{darkviolet}{2402}}^2-267*{\color{darkviolet}{147}}^2\mathbf{\color{blue}{\;=\;}}1\)

(Pell equation solver)

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

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

Splitst men deze periode van \(44\) cijfers in twee gelijke groepen van \(22\) cijfers dan is de som gelijk aan repdigit

\(0037453183520599250936+3295880149812734082397\mathbf{\color{blue}{\;=\;}}{\color{indigo}{3333333333333333333333}}\)

⏮

⯬

⏴

⏵

⯮

⏭