Finding Perfect numbers which are sum of consecutive prime number.

Question

Find all Perfect numbers which are sum of consecutive prime number. For ex . we can write $28$ as : $$28 = 2+3+5+7+11$$

Are there any more examples possible ? If yes , what is the general condition to find such numbers ?

Answer 1

$496$ has a representation : $$5+7+11+\cdots +53+59+61=496$$ For $8128$ and $33550336$ , there is no such representation. Hard to say what is the case for larger perfect numbers.

Answer 2

Some further calculations to add to Peter's answer:

$8589869056$ is the sum of the $1390$ consecutive primes from $6168691$ to $6190871$.

$137438691328$ is the sum of the $112240$ consecutive primes from $454969$ to $2019869$.

$2305843008139952128$ is the sum of the $26$ primes from $88686269543843669$ to $88686269543844787$.

Despite these positive results, there's no obvious pattern and so there's insufficient evidence to say whether perfect numbers are any more or less likely to be sums of consecutive primes than any other positive integers.