\(396\mathbf{\color{blue}{\;=\;}}\) als som van opeenvolgende gehele getallen op vijf verschillende wijzen :

\begin{cases} 396\mathbf{\color{blue}{\;=\;}}5+6+7+8+9+10+11+12+\cdots+20+21+22+23+24+25+26+27+28\\ 396\mathbf{\color{blue}{\;=\;}}31+32+33+34+35+36+37+38+39+40+41\\ 396\mathbf{\color{blue}{\;=\;}}40+41+42+43+44+45+46+47+48\\ 396\mathbf{\color{blue}{\;=\;}}46+47+48+49+50+51+52+53\\ 396\mathbf{\color{blue}{\;=\;}}131+132+133 \end{cases}

\(396\mathbf{\color{blue}{\;=\;}}\) als som van opeenvolgende pare getallen op vijf verschillende wijzen :

\begin{cases} 396\mathbf{\color{blue}{\;=\;}}22+24+26+28+30+32+34+36+38+40+42+44\\ 396\mathbf{\color{blue}{\;=\;}}26+28+30+32+34+36+38+40+42+44+46\\ 396\mathbf{\color{blue}{\;=\;}}36+38+40+42+44+46+48+50+52\\ 396\mathbf{\color{blue}{\;=\;}}96+98+100+102\\ 396\mathbf{\color{blue}{\;=\;}}130+132+134 \end{cases}

\(396\mathbf{\color{blue}{\;=\;}}\) als som van opeenvolgende onpare getallen op drie verschillende wijzen :

\begin{cases} 396\mathbf{\color{blue}{\;=\;}}5+7+9+11+13+15+17+19+21+23+25+27+29+31+33+35+37+39\\ 396\mathbf{\color{blue}{\;=\;}}61+63+65+67+69+71\\ 396\mathbf{\color{blue}{\;=\;}}197+199 \end{cases}

\(396\mathbf{\color{blue}{\;=\;}}197+199\) (som van opeenvolgende priemgetallen)

\(396\mathbf{\color{blue}{\;=\;}}11*(1+2+3+4+5+6+7+8)\) (elf maal de som van opeenvolgende getallen)

\(396\mathbf{\color{blue}{\;=\;}}((0;2;14;14)\,(0;6;6;18)\,(0;10;10;14)\,(1;1;13;15)\,(1;3;5;19)\,(1;5;9;17)\,(1;7;11;15)\)

\(\qquad~~~~(2;2;8;18)\,(2;6;10;16)\,(3;7;7;17)\,(3;7;13;13)\,(3;9;9;15)\,(5;5;11;15)\,(5;9;11;13)\)

\(\qquad~~~~(6;8;10;14))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#15\}\)

\(396\mathbf{\color{blue}{\;=\;}}((0;0;0;0;1;3;3;5;6)\,(0;0;1;1;2;2;2;3;7)\,(0;0;1;3;3;3;4;5;5)\,(0;0;2;3;3;3;3;4;6)\)

\(\qquad~~~~(0;1;1;2;2;4;4;5;5)\,(0;1;2;2;2;3;4;4;6)\,(2;3;3;3;3;3;4;4;5))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#7\}\)

\(396\mathbf{\color{blue}{\;=\;}}2^3+3^3+19^2\)

\(396\mathbf{\color{blue}{\;=\;}}eulerphi(397,437,469,597,603,621,794,796,874,938,1194,1206,1242)\)

\(396\mathbf{\color{blue}{\;=\;}}({\color{blue}{3}}+1)*({\color{blue}{9}}+2)*({\color{blue}{6}}+3)\)

\(396\mathbf{\color{blue}{\;=\;}}tribonacci_{(0,0,9)}[10]\mathbf{\color{blue}{\;=\;}}63+117+216\)

\(396\mathbf{\color{blue}{\;=\;}}tetranacci_{(0,0,5,9)}[10]\mathbf{\color{blue}{\;=\;}}28+56+107+205\)

\(396\mathbf{\color{blue}{\;=\;}}[6^4][36^2]-30^2\mathbf{\color{blue}{\;=\;}}[10^4][100^2]-98^2\mathbf{\color{blue}{\;=\;}}20^2-2^2\mathbf{\color{blue}{\;=\;}}586^2-70^3\)

\(396\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~12\) oplossingen bekend

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

\(396\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

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

\(\qquad~~~~\bbox[3px,border:1px blue solid]{(-45)^5+231^5+(-652)^5+(-689)^5+771^5}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px blue solid]{(-448)^5+1100^5+1912^5+2955^5+(-3023)^5}\)

\(396^2\mathbf{\color{blue}{\;=\;}}404^2-80^2\mathbf{\color{blue}{\;=\;}}429^2-165^2\mathbf{\color{blue}{\;=\;}}445^2-203^2\mathbf{\color{blue}{\;=\;}}471^2-255^2\mathbf{\color{blue}{\;=\;}}495^2-297^2\mathbf{\color{blue}{\;=\;}}565^2-403^2\mathbf{\color{blue}{\;=\;}}660^2-528^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;780^2-672^2\mathbf{\color{blue}{\;=\;}}935^2-847^2\mathbf{\color{blue}{\;=\;}}1125^2-1053^2\mathbf{\color{blue}{\;=\;}}1221^2-1155^2\mathbf{\color{blue}{\;=\;}}1479^2-1425^2\mathbf{\color{blue}{\;=\;}}1804^2-1760^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;2196^2-2160^2\mathbf{\color{blue}{\;=\;}}3279^2-3255^2\mathbf{\color{blue}{\;=\;}}3575^2-3553^2\mathbf{\color{blue}{\;=\;}}4365^2-4347^2\mathbf{\color{blue}{\;=\;}}6540^2-6528^2\mathbf{\color{blue}{\;=\;}}9805^2-9797^2\)

\(396^3\mathbf{\color{blue}{\;=\;}}7881^2-105^2\mathbf{\color{blue}{\;=\;}}7920^2-792^2\mathbf{\color{blue}{\;=\;}}8019^2-1485^2\mathbf{\color{blue}{\;=\;}}8240^2-2408^2\mathbf{\color{blue}{\;=\;}}8250^2-2442^2\mathbf{\color{blue}{\;=\;}}8481^2-3135^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;8494^2-3170^2\mathbf{\color{blue}{\;=\;}}8910^2-4158^2\mathbf{\color{blue}{\;=\;}}9306^2-4950^2\mathbf{\color{blue}{\;=\;}}9930^2-6042^2\mathbf{\color{blue}{\;=\;}}9955^2-6083^2\mathbf{\color{blue}{\;=\;}}\cdots\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;\bbox[2px,border:1px brown dashed]{78606^2-78210^2}\)

Er zijn drie rechthoekige driehoeken met omtrek \(396\) en gehele zijden : \((33;180;183),(72;154;170)\) en \((99;132;165)\)

\(396^2\mathbf{\color{blue}{\;=\;}}4^2-85^2+405^2\mathbf{\color{blue}{\;=\;}}72^2+216^2+324^2\mathbf{\color{blue}{\;=\;}}666^2+804^2-966^2\mathbf{\color{blue}{\;=\;}}\cdots\)

\(396^3\mathbf{\color{blue}{\;=\;}}44^3+264^3+352^3\mathbf{\color{blue}{\;=\;}}-33^3-407^3+506^3\mathbf{\color{blue}{\;=\;}}-918^3-972^3+1206^3\mathbf{\color{blue}{\;=\;}}\cdots~~\) (drie van vele oplossingen)

Als som met de vier operatoren \(+-*\;/\)
\(396\mathbf{\color{blue}{\;=\;}}(55+5)+(55-5)+(55*5)+(55/5)\)
\(396\mathbf{\color{blue}{\;=\;}}(88+2)+(88-2)+(88*2)+(88/2)\)
\(396\mathbf{\color{blue}{\;=\;}}(99+1)+(99-1)+(99*1)+(99/1)\)

\(396\) is het kleinste getal dat gelijk is aan \(22\) maal de som van zijn cijfers : \(396\mathbf{\color{blue}{\;=\;}}22*(3+9+6)\)
Andere getallen met dezelfde eigenschap zijn \(132\) en \(264??~~\) (OEIS A005349 - Harshad getallen)

\(\begin{aligned}396\mathbf{\color{blue}{\;=\;}}\left({\frac{46789273}{5074314}}\right)^3-\left({\frac{37009657}{5074314}}\right)^3\end{aligned}\)

(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)

\(b\mathbf{\color{blue}{\;=\;}}396\to\)
\(b\)\(^{4}\)\(+b\)\(^{8}\)\(+b\)\(^{1}\)\(+b\)\(^{3}\)\(+b\)\(^{6}\)\(+b\)\(^{5}\)\(+b\)\(^{0}\)\(+b\)\(^{5}\)\(+b\)\(^{8}\)\(+b\)\(^{2}\)\(+b\)\(^{5}\)\(+b\)\(^{2}\)\(+b\)\(^{0}\)\(+b\)\(^{9}\)\(+b\)\(^{6}\)\(+b\)\(^{0}\)\(+b\)\(^{1}\)\(+b\)\(^{9}\)\(+b\)\(^{0}\)\(+b\)\(^{4}\)\(+b\)\(^{5}\)\(+b\)\(^{8}\)\(+b\)\(^{8}\)\(+b\)\(^{4}\)\(\mathbf{\color{blue}{\;=\;}}\)
\(481365058252096019045884~~\) (OEIS A236067)

 ○–○–○ 

Som Der Cijfers (\(sdc\)) van \(k^{\large{396}}\) is gelijk aan het grondtal \(k\). De triviale oplossingen \(0\) en \(1\) negerend vinden we :

\(\qquad\qquad~sdc\left(6831^{\large{396}}\right)\mathbf{\color{blue}{\;=\;}}6831\qquad\qquad~sdc\left(6912^{\large{396}}\right)\mathbf{\color{blue}{\;=\;}}6912\)

Expressie met tweemaal de cijfers uit het getal \(396\) enkel met operatoren \(+,-,*,/,(),\)^^
\(396\mathbf{\color{blue}{\;=\;}}(3\)^^\(9\)^^\(6)-3+9-6\)

Als expressie met enkelcijferige toepassing, resp. van \(1\) tot \(9~~\) (met o.a. dank aan Inder. J. Taneja).
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}(1+1+1)*(11+1)*11\)
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}22*(2^{(2+2)}+2)\)
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}33*(3*3+3)\)
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}44*(4+4)+44\mathbf{\color{blue}{\;=\;}}444-44-4\)
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}5*(5*5+55)-5+5/5\)
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}6*66\)
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}7*(7*7+7)-7+77/7\)
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}(8+8)*(8+8+8)+(88+8)/8\)
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}(9+9)*(99+99)/9\)

Met de cijfers van \(1\) tot \(9\) in stijgende en dalende volgorde (met dank aan Inder. J. Taneja) :
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}1+234+5+67+89\)
\(\qquad\qquad396\mathbf{\color{blue}{\;=\;}}9+8*7+6*54+3*2+1\)

\(396*10\)\(^{396}\)\(-1~~\) is een veralgemeend Woodall priemgetal, de dertiende in zijn soort (\(k*10^k-1\)).
(OEIS A059671) & (OEIS A216346)

Kleinste oplossing voor de positieve Pell vergelijking \(x^2-D*y^2\mathbf{\color{blue}{\;=\;}}1~\) met \(D\mathbf{\color{blue}{\;=\;}}396\).

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

\({\qquad\color{darkviolet}{199}}^2-396*{\color{darkviolet}{10}}^2\mathbf{\color{blue}{\;=\;}}1\)

(Pell equation solver)

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

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

(vijf multigrades) \(396\to396^5\to\)

\begin{aligned} 396^1&\mathbf{\color{blue}{\;=\;}}78^1-108^1-264^1+306^1+384^1\\ 396^5&\mathbf{\color{blue}{\;=\;}}78^5-108^5-264^5+306^5+384^5\\ \\ 396^1&\mathbf{\color{blue}{\;=\;}}-44^1-52^1-148^1+252^1+388^1\\ 396^5&\mathbf{\color{blue}{\;=\;}}-44^5-52^5-148^5+252^5+388^5\\ \\ 396^1&\mathbf{\color{blue}{\;=\;}}117^1-162^1+459^1+576^1-594^1\\ 396^5&\mathbf{\color{blue}{\;=\;}}117^5-162^5+459^5+576^5-594^5\\ \\ 396^1&\mathbf{\color{blue}{\;=\;}}-59^1+110^1+342^1-760^1+763^1\\ 396^5&\mathbf{\color{blue}{\;=\;}}-59^5+110^5+342^5-760^5+763^5\\ \\ 396^1&\mathbf{\color{blue}{\;=\;}}201^1-329^1+538^1+814^1-828^1\\ 396^5&\mathbf{\color{blue}{\;=\;}}201^5-329^5+538^5+814^5-828^5\\ \end{aligned}

\(396\) is een getal gelijk aan de som van permutaties, twee aan twee, van deelverzamelingen van zijn eigen cijfers.

\(\qquad396\mathbf{\color{blue}{\;=\;}}36+63+39+93+69+96\)

(Digit-reassembly numbers or Osiris numbers) 132.28 264.20

⏮

⯬

⏴

⏵

⯮

⏭