If router is off for a few days, do you have an ip address as far as ISP is concerned?

This depends on your ISP's DHCP Server.

Each time you are leased a new IP address - it will be leased for a set perios of time (with my ISP, this is 8 days).

Half way through that period of time (4 days in my case), the router will re-apply for its IP and be leased it for a further 8 days - this is known as the DHCP Half-life.

If your router is off long enough that the full length of the DHCP lease is reached and your router makes no attempt to re-lease the IP - this IP address is freed up to go to another client and you lose the IP.

You may find after a long period of time that you power on your router and still get the same IP.

Some DHCP servers will exhaust their list of "never used" IPs before moving onto ones that have been leased and released.. meaning it may not have been reassigned. When your router fires back up, it will ask if it is still allowed to use the same IP (which it thinks it still owns) and be told yes (because nothing else has needed it yet).

Another possibility is that you have a reserved IP. Some ISPs offer static IPs, but still insist that you keep your router set to DHCP. In this case, they set a reservation up - which means that the IP they assign you is reserved exclusively for the MAC address of your router - and it will be "assigned" to you every time your router asks for an IP, no matter how long it has been off for.