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This is follow up to my previous MO question:

A discrete optimization problem related to the AM-GM inequality

Let $n,k$ be integers such that $1\le k\le n$. Define the quantity $$ P(n,k):=\max\ a_1\cdots a_k\ , $$ where the maximum is taken over $k$-tuples of integers $(a_1,\ldots,a_k)$ which are $\ge 1$ and which sum up to $n$.

One can write an explicit formula $$ P(n,k)=\left(\left\lfloor\frac{n}{k}\right\rfloor\right)^{k-n+\left\lfloor\frac{n}{k}\right\rfloor} \left(\left\lfloor\frac{n}{k}\right\rfloor+1\right)^{n-k\left\lfloor\frac{n}{k}\right\rfloor}\ . $$

I would like to learn as much as possible about what is known in the literature regarding the quantity $P(n,k)$. For instance, in the other question MO question, a connection to the Turán graph was mentioned. Any references, which study $P(n,k)$ or where that quantity plays a significant role, would be appreciated. I tried to do a search on the web, but I most got links to finance articles about portfolio maximization which did not quite suit my needs.

This is a follow-up to my previous MO question:

A discrete optimization problem related to the AM-GM inequality

Let $n,k$ be integers such that $1\le k\le n$. Define the quantity $$ P(n,k):=\max\ a_1\cdots a_k\ , $$ where the maximum is taken over $k$-tuples of integers $(a_1,\ldots,a_k)$ which are $\ge 1$ and which sum up to $n$.

One can write the explicit formula $$ P(n,k)=\left(\left\lfloor\frac{n}{k}\right\rfloor\right)^{k-n+\left\lfloor\frac{n}{k}\right\rfloor} \left(\left\lfloor\frac{n}{k}\right\rfloor+1\right)^{n-k\left\lfloor\frac{n}{k}\right\rfloor}\ . $$

I would like to learn as much as possible about what is known in the literature regarding the quantity $P(n,k)$. For instance, in the other MO question, a connection to the Turán graph was mentioned. Any references, which study $P(n,k)$ or where that quantity plays a significant role, would be appreciated. I tried to do a search on the web, but I mostly got links to finance articles about portfolio maximization which did not quite suit my needs.

This is follow up to my previous MO question:

A discrete optimization problem related to the AM-GM inequality

Let $n,k$ be integers such that $1\le k\le n$. Define the quantity $$ P(n,k):=\max a_1\cdots a_k $$ where the maximum is taken over $k$-tuples of integers $(a_1,\ldots,a_k)$ which are $\ge 1$ and which sum up to $n$.

One can write an explicit formula $$ P(n,k)=\left(\left\lfloor\frac{n}{k}\right\rfloor\right)^{k-n+\left\lfloor\frac{n}{k}\right\rfloor} \left(\left\lfloor\frac{n}{k}\right\rfloor+1\right)^{n-k\left\lfloor\frac{n}{k}\right\rfloor}\ . $$

I would like learn as much as possible about what is known in the literature regarding the quantity $P(n,k)$. For instance, in the other question MO question, a connection to the Turán graph was mentioned. Any references, which studies $P(n,k)$ or where that quantity plays a significant role, would be appreciated. I tried to do a search on the web, but I most got links to finance articles about portfolio maximization which did not quite suit my needs.

This is follow up to my previous MO question:

A discrete optimization problem related to the AM-GM inequality

Let $n,k$ be integers such that $1\le k\le n$. Define the quantity $$ P(n,k):=\max\ a_1\cdots a_k\ , $$ where the maximum is taken over $k$-tuples of integers $(a_1,\ldots,a_k)$ which are $\ge 1$ and which sum up to $n$.

One can write an explicit formula $$ P(n,k)=\left(\left\lfloor\frac{n}{k}\right\rfloor\right)^{k-n+\left\lfloor\frac{n}{k}\right\rfloor} \left(\left\lfloor\frac{n}{k}\right\rfloor+1\right)^{n-k\left\lfloor\frac{n}{k}\right\rfloor}\ . $$

I would like to learn as much as possible about what is known in the literature regarding the quantity $P(n,k)$. For instance, in the other question MO question, a connection to the Turán graph was mentioned. Any references, which study $P(n,k)$ or where that quantity plays a significant role, would be appreciated. I tried to do a search on the web, but I most got links to finance articles about portfolio maximization which did not quite suit my needs.