I'm looking for the name of the family of algorithms whose exact runtime is dependent only on the size of the input. An example of such an algorithm would be naive $O(n^3)$ matrix multiplication. If we know $n$, we know exactly how many operations it will execute and how many cycles it will take to complete (ignore real world things like cache misses, context switching, etc.). An algorithm that would not be in that family is Quick sort ($O(n^2)$) since the number of operations is dependent on the contents of the list being sorted. Divider algorithms that only depend on the number of bits of the operands would be in this family, but Euclid's algorithm would not because it finishes faster for some numbers than others.
In other words, suppose you have the algorithm implemented in assembler and the cpu runs one instruction per cycle. Then the value of the program counter $pc(t, n)$ is a function of time $t$ and problem size $n$. The total number of cycles to compute the solution is known in advance.
Though most google hits for "oblivious algorithms" are for "cache-oblivious algorithms"I know:) Not that you mention it, I'm not actually sure if "oblivious algorithms" is a standard name. I'm only sure that "oblivious TM" is a standard name: en.wikipedia.org/wiki/… $\endgroup$