parmenides51 wrote: A positive integer is called lonely if the sum of the inverses of its positive divisors (including $1$ and itself) is not equal with the some of the inverses of the positive divisors of any other positive integer. a) Show that any prime number is lonely. b) Prove that there are infinitely many numbers that are not lonely b) $n=6p$ and $n=28p$, $p:prime>30$

$a)$ is easy, $b)$ is a construction only. $a)$: Assume otherwise. There exists some prime number $p$ and $n$ natural number such that $\frac{1}{p}=\frac{1}{d_1}+\frac{1}{d_2}+...+\frac{1}{d_k}$ From this it is easy to see that $p$ divides some divisor of $n$, but this means $p$ is a divisor of $n$ which makes the RHS larger than LHS.

Jjesus wrote: parmenides51 wrote: A positive integer is called lonely if the sum of the inverses of its positive divisors (including $1$ and itself) is not equal with the some of the inverses of the positive divisors of any other positive integer. a) Show that any prime number is lonely. b) Prove that there are infinitely many numbers that are not lonely b) $n=6p$ and $n=28p$, $p:prime>30$ WHY ARE U SAYING THAT THE PRIME P HAS TO BE GREATER THANK 30 WHAT IS THE PROBLEM WITH P GREATER THAN 7??