What NP-complete problems are most similar to integer factoring?
The textbook of Computational Complexity: Modern Approach, by Arora and Barak gives such example: 

They define the decision problem Integer Factoring on input of three positive integers,

$\text{Integer Factoring} = \{\langle L, U, N \rangle \;|\; (\exists \text{ a prime } p \in \{L, \ldots, U\})[p | N]\}$.

They state that Alon and Kilian showed that if integer $p$ is not required to be prime number then the problem becomes $NP$-complete.