You could set up an alarm contact (using additional hardware which is not part of the generator) that would trigger a phone dialer at whatever time delay you wanted.
At this point we just have a phone line... and with no power for a couple of day's the pipes would freeze... in january, it could even happen in less than a day... Like I said we go there every weekend, so to drain the heating system and all pipes every weekend really is not an option, this is why i got the generator... I know there are power outage alert systems that call preprogramed numbers, but would they call you if a generator kicks in 30 seconds after the outage? If i knew the generator was running from my primary home, I could monitor the time on, and if need be drive up.. or call the electric co. to find out how long the outage is expected to last... Last year they had a 4 day outage...We did not own the house then..
I would probably have the generator's output voltage activate a relay which then triggers whatever you use for the phone dialer. If you use an additional relay which is activated by utility power and wire the contacts of the two relays appropriately, then you could avoid a phone call when the generator does its weekly test run.