\(160\mathbf{\color{blue}{\;=\;}}30+31+32+33+34\) (som van opeenvolgende gehele getallen)

\(160\mathbf{\color{blue}{\;=\;}}28+30+32+34+36\) (som van opeenvolgende pare getallen)

\(160\mathbf{\color{blue}{\;=\;}}7+9+11+13+15+17+19+21+23+25\mathbf{\color{blue}{\;=\;}}13+15+17+19+21+23+25+27\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~37+39+41+43\mathbf{\color{blue}{\;=\;}}79+81\) (som van opeenvolgende onpare getallen)

\(160\mathbf{\color{blue}{\;=\;}}2+3+5+7+11+13+17+19+23+29+31\) (som van opeenvolgende priemgetallen)

\(160\mathbf{\color{blue}{\;=\;}}((0;0;4;12)\,(4;4;8;8))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#2\}\)

\(160\mathbf{\color{blue}{\;=\;}}(1^2+2^2)*(4^2+4^2)\mathbf{\color{blue}{\;=\;}}(2^2+2^2)*(2^2+4^2)\)

\(160\mathbf{\color{blue}{\;=\;}}\bbox[#f4f4f4,3px,border:1px solid]{2^3+3^3+5^3}\mathbf{\color{blue}{\;=\;}}((0;0;0;0;0;0;2;3;5)\,(0;0;0;2;2;2;2;4;4)\)

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

\(160\mathbf{\color{blue}{\;=\;}}2^5+2^6+2^6\)

\(160\mathbf{\color{blue}{\;=\;}}2^5+3^1+5^3\mathbf{\color{blue}{\;=\;}}2^7+3^3+5^1\)

\(160\mathbf{\color{blue}{\;=\;}}eulerphi(187,205,328,352,374,400,410,440,492,528,600,660)\)

\(160\mathbf{\color{blue}{\;=\;}}32*(3+2)~~\) (\(32\) is een deler van \(160\))

\(160\mathbf{\color{blue}{\;=\;}}[2^4][4^2]+12^2\mathbf{\color{blue}{\;=\;}}2^5+2^7\mathbf{\color{blue}{\;=\;}}13^2-3^2\mathbf{\color{blue}{\;=\;}}14^2-6^2\mathbf{\color{blue}{\;=\;}}22^2-18^2\mathbf{\color{blue}{\;=\;}}41^2-39^2\)

\(160\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~37\) oplossingen bekend

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

\(160\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

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

\(\qquad~~~~\bbox[3px,border:1px blue solid]{2^5+2^5+2^5+2^5+2^5}\mathbf{\color{blue}{\;=\;}}\)

\(160\mathbf{\color{blue}{\;=\;}}\)(som van zeven zevendemachten)

\(\qquad~~~~\bbox[3px,border:1px blue solid]{(-32)^7+(-68)^7+85^7+(-104)^7+(-112)^7+(-117)^7+130^7}\)

\(160^2\mathbf{\color{blue}{\;=\;}}[2^{14}][4^7][128^2]+96^2\mathbf{\color{blue}{\;=\;}}164^2-[6^4][36^2]\mathbf{\color{blue}{\;=\;}}178^2-78^2\mathbf{\color{blue}{\;=\;}}200^2-120^2\mathbf{\color{blue}{\;=\;}}232^2-168^2\mathbf{\color{blue}{\;=\;}}281^2-231^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;340^2-300^2\mathbf{\color{blue}{\;=\;}}416^2-384^2\mathbf{\color{blue}{\;=\;}}650^2-630^2\mathbf{\color{blue}{\;=\;}}808^2-792^2\mathbf{\color{blue}{\;=\;}}1285^2-1275^2\mathbf{\color{blue}{\;=\;}}1604^2-1596^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;3202^2-3198^2\mathbf{\color{blue}{\;=\;}}6401^2-6399^2\)

\(160^3\mathbf{\color{blue}{\;=\;}}96^3+1792^2\mathbf{\color{blue}{\;=\;}}640^2+1920^2\mathbf{\color{blue}{\;=\;}}1152^2+1664^2\mathbf{\color{blue}{\;=\;}}1664^2+1152^2\mathbf{\color{blue}{\;=\;}}1792^2+96^3\mathbf{\color{blue}{\;=\;}}1920^2+640^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;2024^2-24^2\mathbf{\color{blue}{\;=\;}}2080^2-480^2\mathbf{\color{blue}{\;=\;}}2240^2-960^2\mathbf{\color{blue}{\;=\;}}2512^2-1488^2\mathbf{\color{blue}{\;=\;}}2548^2-1548^2\mathbf{\color{blue}{\;=\;}}2960^2-2160^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;3520^2-2880^2\mathbf{\color{blue}{\;=\;}}4256^2-3744^2\mathbf{\color{blue}{\;=\;}}4346^2-3846^2\mathbf{\color{blue}{\;=\;}}5320^2-4920^2\mathbf{\color{blue}{\;=\;}}6560^2-6240^2\mathbf{\color{blue}{\;=\;}}8128^2-7872^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;8317^2-8067^2\mathbf{\color{blue}{\;=\;}}\ldots\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{12880^2-12720^2}\)

Vertrekkend van \(160\) en telkens de som makend van de derde machten van de cijfers maken we de cirkel rond:

\(\underline{160}\to1^3+6^3+0^3\mathbf{\color{blue}{\;=\;}}217\to2^3+1^3+7^3\mathbf{\color{blue}{\;=\;}}352\to3^3+5^3+2^3\mathbf{\color{blue}{\;=\;}}\underline{160}\)

In het Engels worden deze RDI's genoemd wat staat voor Recurring Digital Invariants.

Zie ook bij 55.9 136.4 919.--

Drie kwadraten met dezelfde cijfers : \({\color{blue}{160}}^2\mathbf{\color{blue}{\;=\;}}25600~; 245^2\mathbf{\color{blue}{\;=\;}}60025~\) en \(~250^2\mathbf{\color{blue}{\;=\;}}62500\)

Als som met de vier operatoren \(+-*\;/\)
\(160\mathbf{\color{blue}{\;=\;}}(30+3)+(30-3)+(30*3)+(30/3)\)
\(160\mathbf{\color{blue}{\;=\;}}(40+1)+(40-1)+(40*1)+(40/1)\)

\(\begin{align}160\mathbf{\color{blue}{\;=\;}}\left({\frac{2}{7}}\right)^3+\left({\frac{38}{7}}\right)^3\mathbf{\color{blue}{\;=\;}}\color{tomato}{\left({\frac{1*2}{7}}\right)^3+\left({\frac{19*2}{7}}\right)^3\mathbf{\color{blue}{\;=\;}}\left({\frac{1}{7}}\right)^3+\left({\frac{19}{7}}\right)^3\mathbf{\color{blue}{\;=\;}}{\frac{160}{8}}\mathbf{\color{blue}{\;=\;}}20}\end{align}\)

(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)

\(b\mathbf{\color{blue}{\;=\;}}160\to b\)\(^{1}\)\(+b\)\(^{0}\)\(+b\)\(^{7}\)\(+b\)\(^{3}\)\(+b\)\(^{7}\)\(+b\)\(^{4}\)\(+b\)\(^{2}\)\(+b\)\(^{0}\)\(+b\)\(^{2}\)\(+b\)\(^{1}\)\(+b\)\(^{4}\)\(+b\)\(^{3}\)\(+b\)\(^{7}\)\(+b\)\(^{4}\)\(+b\)\(^{7}\)\(+b\)\(^{2}\)\(+b\)\(^{2}\)\(\mathbf{\color{blue}{\;=\;}}10737420214374722~~\)
(OEIS A236067)

Som Der Cijfers (\(sdc\)) van \(k^{\large{160}}\) is gelijk aan het grondtal \(k\). De triviale oplossingen \(0\) en \(1\) negerend vinden we :

\(\qquad\qquad~sdc\left(400^{\large{160}}\right)\mathbf{\color{blue}{\;=\;}}400\quad\qquad\qquad~sdc\left(2473^{\large{160}}\right)\mathbf{\color{blue}{\;=\;}}2473\qquad\qquad~sdc\left(2488^{\large{160}}\right)\mathbf{\color{blue}{\;=\;}}2488\)

\(\qquad\qquad~sdc\left(2574^{\large{160}}\right)\mathbf{\color{blue}{\;=\;}}2574\)

Expressies met tweemaal de cijfers uit het getal \(160\) enkel met operatoren \(+,-,*,/,(),\)^^\(\)
\(160\mathbf{\color{blue}{\;=\;}}(1\)^^\(6\)^^\(0)+1*6*0\mathbf{\color{blue}{\;=\;}}(1\)^^\(6)*(1\)^^\(0)+6*0\)

Als expressie met enkelcijferige toepassing, resp. van \(1\) tot \(9~~\) (met dank aan Inder. J. Taneja).
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}(1+1)*((11-1-1)^{(1+1)}-1)\)
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}2*2*2*(22-2)\)
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}(3+3)*3^3-3+3/3\)
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}4*(44-4)\)
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}5*((5+5)/5)^5\)
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}6*6+6+6+(666+6)/6\)
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}7*7+777/7~~{\small\color{green}{(Origineel{\color{red}{~777+7*7/7~}}is~onjuist)}}\)
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}88+8*8+8\)
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}9*(9+9)-(9+9)/9\)

Met de cijfers van \(1\) tot \(9\) in stijgende en dalende volgorde (met dank aan Inder. J. Taneja) :
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}12+3*4+5+6*7+89\)
\(\qquad\qquad160\mathbf{\color{blue}{\;=\;}}98+7*6+5+4*3+2+1\)

(vier multigrades) \(160\to160^5\to\)

\begin{aligned} 160^1&\mathbf{\color{blue}{\;=\;}}-96^1-140^1+436^1+508^1-548^1\\ 160^5&\mathbf{\color{blue}{\;=\;}}-96^5-140^5+436^5+508^5-548^5\\ \\ 160^1&\mathbf{\color{blue}{\;=\;}}12^1-672^1+860^1+1116^1-1156^1\\ 160^5&\mathbf{\color{blue}{\;=\;}}12^5-672^5+860^5+1116^5-1156^5\\ \\ 160^1&\mathbf{\color{blue}{\;=\;}}420^1+860^1-1240^1-1360^1+1480^1\\ 160^5&\mathbf{\color{blue}{\;=\;}}420^5+860^5-1240^5-1360^5+1480^5\\ \\ 160^1&\mathbf{\color{blue}{\;=\;}}-280^1-1690^1+2330^1+2620^1-2820^1\\ 160^5&\mathbf{\color{blue}{\;=\;}}-280^5-1690^5+2330^5+2620^5-2820^5\\ \end{aligned}

Kleinste oplossing voor de positieve Pell vergelijking \(x^2-D*y^2\mathbf{\color{blue}{\;=\;}}1~\) met \(D\mathbf{\color{blue}{\;=\;}}160\).

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

\(\qquad{\color{darkviolet}{721}}^2-160*{\color{darkviolet}{57}}^2\mathbf{\color{blue}{\;=\;}}1\)

(Pell equation solver)

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