If I understand it well now, what you want are books about **basic** number theory with a good algebraic foundation. I can recommend the following:

* *Elementary methods in number theory* (2000), by **Nathanson**. Graduate Texts in Mathematics. It starts low, but it reaches quite high.

* *Elementary Number Theory with Applications* (2007) by **Koshy**. Elsevier. Truly basic. Not very very algebraic, but a really nice textbook.

* *Algebra and number theory* by Andrew **Baker**. Online notes. Fairly basic.

* *A computational introduction to number theory and algebra* (2005) by **Shoup**. Cambridge University Press / Online Free Version. Despite the title, I think it satisfies the conditions you are looking for.