Write a bijection to match the set of positive integers with the set of positive integers excluding the multiples of 4.

Question

Write a bijection to match the set of positive integers with the set of positive integers excluding the multiples of 4.

e.g. $$1\mapsto1,2\mapsto2, 3\mapsto3, 4\mapsto 5, 5\mapsto , 6\mapsto 7, 7\mapsto9, \dots $$

I don't have any idea where to start the process of defining such a bijection. And is it possible to define more than one bijection for the particular question above?

Answer 1

For example you can try $$f(n)=n + \lfloor (n-1)/3\rfloor.$$ Does it work? Are you able to find another bijection?

As a bonus question, find a bijection from the set of of positive integers with the set of positive integers excluding the multiples of a given integer $m>1$.