Your money is lost without at least one of
a backup of important data from the hard disk before it was formatted
a backup of the wallet.dat file
a note of the private key
There is no way to recover the private key from other information. If there was, Bitcoin would be completely unsafe and unusable.
I'm afraid that if you only have the addresses and not the corresponding private keys you're out of luck here.
To recover a wallet or an address you need to have either the wallet.dat or the private key to a Bitcoin address.
The password you made back then was likely used to encrypt the wallet.dat.
I solved the problem by installing the bitcoin package again.
I was getting this error message with my script doubleslow-base.py:
Traceback (most recent call last):
File "./doubleslow-base.py", line 136, in <module>
File "/home/valentin/python/doubleslow/doubleslow_module.py", line 301, in ...
There are various ways to extract and dump the private/public keys from a wallet.dat depending on how it became corrupted.
First, try to run -salvagewallet command to repair it.
If you have a tool to dump the mongo db, yes its possible to find possible keypairs (pub/priv) and dump them out using python tools. Of course you would need the passphrase for this.