A positive integer is called lonely if the sum of the reciprocals of its positive divisors (including 1 and itself) is different from the sum of the reciprocals of the positive divisors of any positive integer. a) Prove that every prime number is lonely. b) Prove that there are infinitely many positive integers that are not lonely.
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Tags: number theory
FabrizioFelen
30.10.2015 06:30
My solution: Part a) Let $S(n)$ the sum of divisors of $n$ and $T(n)$ the sum of the reciprocal divisors of $n$ $\Longrightarrow$ $T(n)=\frac{S(n)}{n}$ Suposse that exists $n$ such that $T(n)=T(p)$ where $p$ is prime $\Longrightarrow$ $\frac{S(n)}{n}=\frac{S(p)}{p}$ $\Longrightarrow$ $p|n$ and $p\neq n$ and $n=pk$ $\Longrightarrow$ $S(n)\geq n+k+1$ $\Longrightarrow$ $T(n)\geq \frac{pk+k+1}{pk}> \frac{p+1}{p}=T(p)$ $\Longrightarrow$ $T(n)\neq T(p)$ $\Longrightarrow$ $p$ is lonely for all prime $p$ Part b) We know that: $T(mn)=T(n).T(m)$ for all $m,n$ coprime $\Longrightarrow$ $T(n)$ is multiplicative It is easy to see that: $T(6)=2$ and $T(28)=2$ $\Longrightarrow$ $T(6k)=2T(k)=T(28k)$ for all $k$ coprime to $42$ $\Longrightarrow$ $n=6k$ is not lonely for all $k$ coprime to $42$ $\Longrightarrow$ exists infinitely integers $n=6k$ that are not lonely...