论文标题
关于$ n $的代表数量，$ n $的奇数除数的倒数总和作为$ r $ quares的总和
An identity for the sum of inverses of odd divisors of $n$ in terms of the number of representations of $n$ as a sum of $r$ squares
论文作者
论文摘要
令$ \ sum _ {\替换{d | n \\ d \ equiv 1（2）}} \ frac {1} {d} {d} $$表示正整数$ n $的奇数分隔的倒数总数符号被视为独特。该注释的目的是证明以下有趣的组合身份：$$ \ sum _ {\ ordack {d | n \\ d \ equiv 1 （2）}}} \ frac {1} {d} = \ frac {1} {2} \，\ sum_ {r = 1}^{n} \ frac {（ - 1）^{n+r}}}}} {n+r}} {r} {r} {r} {r} {r} {r} $$
Let $$\sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}$$ denote the sum of inverses of odd divisors of a positive integer $n$, and let $c_{r}(n)$ be the number of representations of $n$ as a sum of $r$ squares where representations with different orders and different signs are counted as distinct. The aim is of this note is to prove the following interesting combinatorial identity: $$ \sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}=\frac{1}{2}\,\sum_{r=1}^{n}\frac{(-1)^{n+r}}{r}\,\binom{n}{r}\, c_{r}(n). $$
