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For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since $$6=1+1+4=1+2+3=2+2+2.$$ QUESTION. ...

Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers). Let $$ \ell(n)=\left\lfloor\log_2 n\right\rfloor $$ Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...

For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$. For instance: $$ 100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...

Let $\operatorname{tr}(n)$ be A007814, number of trailing zeros in the binary representation of $n$. Also, let $\operatorname{ntr}(n)$ be A086784, number of non-trailing zeros in the binary ...

I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem: Denote $R(N,L)$ ...