Mitigating unknown error in power analysis of ATMega328P microcontroller
My suspicion would be that it has nothing to do with the instructions at all. Something else in the microcontroller is becoming active every 8 and/or 16 clock cycles.

A couple of possibilities that come to mind are:


A timer/counter peripheral has its clock prescaler set to 8.
One of the timer/counter peripherals is rolling over every 8 or 16 clock cycles.
The SPI peripheral has its clock prescaler set to 16.
The TWI (aka. I2C) peripheral has its clock prescaler set to 16.
The CKDIV8 fuse is programmed.