\(119\mathbf{\color{blue}{\;=\;}}2+3+4+5+6+7+8+9+10+11+12+13+14+15\mathbf{\color{blue}{\;=\;}}14+15+16+17+18+19+20\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~59+60\) (som van opeenvolgende gehele getallen)

\(119\mathbf{\color{blue}{\;=\;}}11+13+15+17+19+21+23\) (som van opeenvolgende onpare getallen)

\(119\mathbf{\color{blue}{\;=\;}}7+11+13+17+19+23+29\mathbf{\color{blue}{\;=\;}}17+19+23+29+31\) (som van opeenvolgende priemgetallen)

\(119\mathbf{\color{blue}{\;=\;}}3+6+10+15+21+28+36\) (som van opeenvolgende driehoeksgetallengetallen)

\(119\mathbf{\color{blue}{\;=\;}}1*1!+2*2!+3*3!+4*4!\mathbf{\color{blue}{\;=\;}}5!-1\)

\(119\mathbf{\color{blue}{\;=\;}}((1;1;6;9)\,(1;3;3;10)\,(2;3;5;9)\,(3;5;6;7))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#4\}\)

\(119\mathbf{\color{blue}{\;=\;}}1^3+3^3+3^3+4^3\mathbf{\color{blue}{\;=\;}}((0;0;0;0;0;1;3;3;4)\,(0;1;1;1;2;3;3;3;3)\,(1;1;1;1;2;2;2;3;4))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#3\}\)

\(119\mathbf{\color{blue}{\;=\;}}11*9+(11+9)\)

\(119\mathbf{\color{blue}{\;=\;}}tribonacci_{(1,9,1)}[8]\mathbf{\color{blue}{\;=\;}}21+33+65\)

\(119\mathbf{\color{blue}{\;=\;}}tetranacci_{(0,0,1,7)}[9]\mathbf{\color{blue}{\;=\;}}8+16+32+63\)

\(119\mathbf{\color{blue}{\;=\;}}2^7-3^2\mathbf{\color{blue}{\;=\;}}12^2-5^2\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{60^2-59^2}\mathbf{\color{blue}{\;=\;}}386^2-53^3\)

\(119\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~18\) oplossingen bekend

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

\(119\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

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

\(\qquad~~~~\)(oplossingen met vijf vijfcijfer_termen door Joe Wetherell)

\(\qquad~~~~\bbox[3px,border:1px blue solid]{(-3)^5+(-30)^5+33^5+41^5+(-42)^5}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px blue solid]{234^5+248^5+(-429)^5+(-499)^5+535^5}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px blue solid]{(-13607)^5+(-31615)^5+32621^5+34556^5+(-35216)^5}\)

\(\qquad~~~~\bbox[3px,border:1px blue solid]{14441^5+(-26602)^5+(-42210)^5+(-43461)^5+49651^5}\)

\(119\mathbf{\color{blue}{\;=\;}}\)(som van zeven zevendemachten)

\(\qquad~~~~\bbox[3px,border:1px red dashed]{0^7+0^7+4^7+(-6)^7+(-6)^7+(-6)^7+7^7}\)

\(119^2\mathbf{\color{blue}{\;=\;}}[13^4][169^2]-120^2\mathbf{\color{blue}{\;=\;}}21^3+70^2\mathbf{\color{blue}{\;=\;}}56^2+105^2\mathbf{\color{blue}{\;=\;}}425^2-408^2\mathbf{\color{blue}{\;=\;}}1015^2-1008^2\mathbf{\color{blue}{\;=\;}}1037^2-102^3\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\,\,7081^2-7080^2\)

\(119^3\mathbf{\color{blue}{\;=\;}}140^3-1029^2\mathbf{\color{blue}{\;=\;}}1428^2-595^2\mathbf{\color{blue}{\;=\;}}2628^2-2285^2\mathbf{\color{blue}{\;=\;}}3060^2-2771^2\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{7140^2-7021^2}\)

De eerste keer dat er \(119\) opeenvolgende samengestelde getallen voorkomen gebeurt tussen de priemgetallen \(1895359\)
en \(1895479\) met aldus een priemkloof van \(120\,.~~\) (OEIS A000101.pdf)

\(b\mathbf{\color{blue}{\;=\;}}119\to b\)\(^{4}\)\(+b\)\(^{8}\)\(+b\)\(^{2}\)\(+b\)\(^{6}\)\(+b\)\(^{0}\)\(+b\)\(^{0}\)\(+b\)\(^{6}\)\(+b\)\(^{0}\)\(+b\)\(^{7}\)\(+b\)\(^{5}\)\(+b\)\(^{5}\)\(+b\)\(^{9}\)\(+b\)\(^{2}\)\(+b\)\(^{3}\)\(+b\)\(^{0}\)\(+b\)\(^{3}\)\(+b\)\(^{1}\)\(+b\)\(^{2}\)\(+b\)\(^{3}\)\(\mathbf{\color{blue}{\;=\;}}4826006075592303123~~\)
(OEIS A236067)

 ○–○–○ 

Som Der Cijfers (\(sdc\)) van \(k^{\large{119}}\) is gelijk aan het grondtal \(k\). De triviale oplossingen \(0\) en \(1\) negerend vinden we :

\(\qquad\qquad~sdc\left(1665^{\large{119}}\right)\mathbf{\color{blue}{\;=\;}}1665\qquad\qquad~sdc\left(1673^{\large{119}}\right)\mathbf{\color{blue}{\;=\;}}1673\)

Expressie met tweemaal de cijfers uit het getal \(119\) bij voorkeur met de basisoperatoren
\(119\mathbf{\color{blue}{\;=\;}}(1\)^^\(1)+(9+(9*(1\)^^\(1)))\mathbf{\color{blue}{\;=\;}}(1\)^^\(1\)^^\(9)*1*(1\)^\(9)\mathbf{\color{blue}{\;=\;}}(1\)^^\(1)*9+(1\)^^\(1)+9\)

Als expressie met enkelcijferige toepassing, resp. van \(1\) tot \(9~~\) (met dank aan Inder. J. Taneja).
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}11*11-1-1\mathbf{\color{blue}{\;=\;}}11^{(1+1)}-1-1\)
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}(22/2)^2-2\)
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}3*3+(333-3)/3\)
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}4+4+444/4\)
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}5*5*5-5-5/5\)
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}6+(666+6+6)/6\)
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}7*7+77-7\)
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}8+888/8\)
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}9+(999-9)/9\)

Met de cijfers van \(1\) tot \(9\) in stijgende en dalende volgorde (met dank aan Inder. J. Taneja) :
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}1+2+3+4*5+6+78+9\)
\(\qquad\qquad119\mathbf{\color{blue}{\;=\;}}9*8+7+6*5+4+3+2+1\)

Som der reciproken van partitiegetallen van \(119\) is \(1\) op \(314\) (driehonderdveertien) wijzen.

Negen partities hebben unieke termen.

(OEIS A125726)

(multigrades) \(119\to1375298099\to\)

\begin{align} 3^1+54^1+62^1&\mathbf{\color{blue}{\;=\;}}24^1+28^1+67^1\\ 3^5+54^5+62^5&\mathbf{\color{blue}{\;=\;}}24^5+28^5+67^5\\ \end{align} (multigrades) \(119\to1375298099~~\text{(\(+\,28^5\) is pannumerisch \(1392508467~\))}\to\)

\begin{align} 3^1+{\color{red}{28}}^1+54^1+62^1&\mathbf{\color{blue}{\;=\;}}24^1+{\color{red}{28}}^1+28^1+67^1\\ 3^5+{\color{red}{28}}^5+54^5+62^5&\mathbf{\color{blue}{\;=\;}}24^5+{\color{red}{28}}^5+28^5+67^5 \end{align}

Kleinste oplossing voor de positieve Pell vergelijking \(x^2-D*y^2\mathbf{\color{blue}{\;=\;}}1~\) met \(D\mathbf{\color{blue}{\;=\;}}119\).

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

\(\qquad{\color{darkviolet}{120}}^2-119*{\color{darkviolet}{11}}^2\mathbf{\color{blue}{\;=\;}}1\)

(Pell equation solver)

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

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

Splitst men deze periode van \(48\) cijfers in twee gelijke groepen van \(24\) cijfers dan is de som gelijk aan

\(008403361344537815126050+420168067226890756302521\mathbf{\color{blue}{\;=\;}}{\color{indigo}{428571\_428571\_428571\_428571}}\)

hetgeen de aaneenschakeling van vier maal de cijfers \(428571\) voorstelt.

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