On Determination of $$\varphi ^{-1}\left( 2^{a} {p}^{{a}_1} \right) $$

In this paper, we have considered the set \(\varphi ^{-1}\left( 2^a p^{a_1} \right) \) where p is an odd prime, \(a, a_1\in \mathbb {N}\) and have shown how to determine the elements of it. Several important results on behalf of this have been discussed.