\(164=17+18+19+20+21+22+23+24\) (som van opeenvolgende gehele getallen)

\(164=38+40+42+44\) (som van opeenvolgende pare getallen)

\(164=81+83+85\) (som van opeenvolgende onpare getallen)

\(164\mathbf{\color{blue}{\;=\;}}3+6+10+15+21+28+36+45\mathbf{\color{blue}{\;=\;}}28+36+45+55\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~D(2)+D(3)+D(4)+D(5)+D(6)+D(7)+D(8)+D(9)\mathbf{\color{blue}{\;=\;}}D(7)+D(8)+D(9)+D(10)\)

\(\qquad~~~~\)(som van opeenvolgende driehoeksgetallen op twee wijzen)

\(164=3^2+5^2+7^2+9^2\) (som van kwadraten van opeenvolgende onpare getallen)

\(164=((0;0;8;10)\,(0;2;4;12)\,(0;6;8;8)\,(1;1;9;9)\,(3;3;5;11)\,(3;5;7;9))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#6\}\)

\(164=((0;0;0;0;1;2;3;4;4)\,(0;0;1;1;1;1;2;3;5)\,(0;1;1;3;3;3;3;3;3)\,(1;1;1;2;2;3;3;3;4))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#4\}\)

\(164=2^4+3^3+11^2\)

\(164\mathbf{\color{blue}{\;=\;}}[2^6][4^3][8^2]+10^2\mathbf{\color{blue}{\;=\;}}2^7+6^2\mathbf{\color{blue}{\;=\;}}14^2-2^5\mathbf{\color{blue}{\;=\;}}17^2-5^3\mathbf{\color{blue}{\;=\;}}26^2-[2^9][8^3]\mathbf{\color{blue}{\;=\;}}42^2-40^2\)

\(164\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~15\) oplossingen bekend

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

\(164\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

\(\qquad~~~~\bbox[3px,border:1px blue solid]{(-3)^5+(-6)^5+(-8)^5+(-9)^5+10^5}\)

\(164^2=45^2+160^2-27^2\)

\(164^2\mathbf{\color{blue}{\;=\;}}[6^4][36^2]+160^2\mathbf{\color{blue}{\;=\;}}41^3-205^2\mathbf{\color{blue}{\;=\;}}58^4-3360^2\mathbf{\color{blue}{\;=\;}}205^2-123^2\mathbf{\color{blue}{\;=\;}}1685^2-1677^2\mathbf{\color{blue}{\;=\;}}3364^2-3360^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;6725^2-6723^2\)

\(164^3\mathbf{\color{blue}{\;=\;}}920^2+1888^2\mathbf{\color{blue}{\;=\;}}1312^2+1640^2\mathbf{\color{blue}{\;=\;}}2337^2-1025^2\mathbf{\color{blue}{\;=\;}}3690^2-3034^2\mathbf{\color{blue}{\;=\;}}6888^2-6560^2\mathbf{\color{blue}{\;=\;}}\cdots\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;\bbox[2px,border:1px brown dashed]{13530^2-13366^2}\)

Als som met de vier operatoren \(+-*\;/\)
\(164=(41+1)+(41-1)+(41*1)+(41/1)\)

Voor \(n=164~~\) geldt \(~~{\large\phi}(n)={\large\phi}(n+1) ~~\to~~ {\large\phi}(164)={\large\phi}(165)=80~~~~({\large\phi}\) of  'phi' staat voor totiënt)

Zie ook bij 15.35 en 104.11   (OEIS A001274)

Voor \(n=164~~\) geldt \(~~{\large\sigma}(n)={\large\sigma}(n+30) ~~\to~~ {\large\sigma}(164)={\large\sigma}(194)=294~~~~({\large\sigma}\) of  'sigma' staat voor som der delers)

\(164\) is de derde oplossing uit de reeks \(88,161,164,209,221,275,279,376,497,581,707,869,910,913,1015,\ldots\).

Zie bvb. bij 88.18

\(\begin{align}164\mathbf{\color{blue}{\;=\;}}\left({\frac{311155001}{46913867}}\right)^3-\left({\frac{236283589}{46913867}}\right)^3\end{align}\)

(Integral Sum of Two Rational Cubes) (OEIS A020898) (OEIS A228499)

\((x^3+y^3)/z^3=n~\to~\) [x waarde] (OEIS A190356)  [y waarde] (OEIS A190580)  [z waarde] (OEIS A190581)

Kleinste positieve oplossingen \(~\to~\) [x waarde] (OEIS A254326)  [y waarde] (OEIS A254324)

\(b\mathbf{\color{blue}{\;=\;}}164\to\)
\(b\)\(^{2}\)\(+b\)\(^{5}\)\(+b\)\(^{7}\)\(+b\)\(^{9}\)\(+b\)\(^{9}\)\(+b\)\(^{3}\)\(+b\)\(^{3}\)\(+b\)\(^{1}\)\(+b\)\(^{1}\)\(+b\)\(^{4}\)\(+b\)\(^{3}\)\(+b\)\(^{5}\)\(+b\)\(^{9}\)\(+b\)\(^{1}\)\(+b\)\(^{1}\)\(+b\)\(^{7}\)\(+b\)\(^{3}\)\(+b\)\(^{1}\)\(+b\)\(^{2}\)\(+b\)\(^{8}\)\(+b\)\(^{4}\)\(\mathbf{\color{blue}{\;=\;}}257993311435911731284~~\)
(OEIS A236067)