Y by centslordm, mathleticguyyy, Mathematicsislovely, RedFlame2112, jhu08, rama1728, megarnie, mathmax12, farhad.fritl, cubres
Determine all functions
defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:

for at least one
;
for every positive integers
and
;
there are infinitely many positive integers
such that
for all
.