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For some ${n \in \mathbb{N}}$ and ${f(x_1, x_2, x_3, x_5,...) = 1^{x_1}.2^{x_2}.3^{x_3}.5^{x_5}.7^{x_7}....}$

${x_i \in \mathbb{W}}$

Find ${x_1, x_2, x_3, x_5,...}$ such that ${f(x_1, x_2, x_3, x_5,...)}$ is maximum and ${1.x_1 + 2.x_2 + 3.x_3 + 5.x_5 + ... = n}$

Sorry if already asked, did not know how to search this...

Akshay Mariyanna
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  • I think you also need $x_i>0$ constraints (or at least $x_1>0$, otherwise I don't think there is a maximum).Are $x_i$ assumed to be $\Bbb N$? Have you tried Kuhn-Tucker on $\ln f(x_1,....)$? Also, is the number of $i$s fixed? – Patricio Jan 17 '19 at 14:55
  • @Patricio - Edited to define bounds of x and n – Akshay Mariyanna Jan 17 '19 at 15:10

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Note that $3\cdot(x-3)>x$ when $x\geq5$. Therefore we should replace any factor $x\geq5$ by the pair $3$, $x-3$. Furthermore $4=2\cdot 2$, and a factor $1$ is of no help. It follows that we should choose all factors $\in\{2,3\}$. In addition note that $2+2+2=3+3$, but $2^3<3^2$. This means that all factors should be $3$, except at most two factors $2$.

From these facts it is not too difficult to arrive at the maximal product for given factor sum $n$.

Christian Blatter
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