The smallest two perfect numbers $n=6$ and $m=28$ satisfy $$ \frac{m}{n+1} = 2^k $$

with $k=2.$

Question: Are there more pairs of perfect numbers $n,m$ with $n < m$ and such that

$$ \frac{m}{n+1} = 2^k $$

for some positive integer $k>0.$

The smallest two perfect numbers $n=6$ and $m=28$ satisfy $$ \frac{m}{n+1} = 2^k $$

with $k=2.$

Question: Are there more pairs of perfect numbers $n,m$ with $n < m$ and such that

$$ \frac{m}{n+1} = 2^k $$

for some positive integer $k>0.$

Observe that the perfect number $n$ , the smallest of $n,m$ may be also an odd number.