powering the arduino uno
The Arduino Uno is not really designed for that. Simply keeping the power indicator LED, regulators and extra circuits on for a week will require in the order of 1500-2000 mAh (10*24*7=1680mAh). That is without any application at all on the MCU. 

Now it is possible to modify the board and reduce this extra power requirements to less than 1 mA and as low as 0.1 mA. Or even better buy a Pro Mini clone and cut the wiring to the LED and regulator, and power with a 3.7 V/1500mAh battery directly. 

Here is a great reference on the subject. 

Cheers!