Remember password after `sudo -i`

Remember password after `sudo -i`

If I do sudo <command> and enter the password, my Debian Squeeze remembers the fact I passed the password and never asks for it for i.e. 15 minutes.
But if I do sudo -i and then exit, the system does not remembers the password and asks for it all over again every time I use sudo -i. Doing just once sudo <command> makes the system not ask for password even for a next sudo -i.
Here are the scenarios I've described above:
First one:
login (by SSH),
sudo <command>: system's asking for the password,
sudo -i: system is not asking for the password.
Second one:
login (by SSH),
sudo -i: system's asking for the password,
exit
sudo -i: system's again asking for the password.
Doing sth beetween 2. and 3. makes the system remember the password.
What is the reason of such a behaviour? Is it possible to make the system remember the password (fact I passed it) also after using sudo -i?