Oracles which put integer factorization in P
(This is shameless self-promotion.) If you don’t mind either assuming the generalized Riemann hypothesis (for $L$-functions of quadratic Dirichlet characters) or using randomized polynomial time, then the following search problems work:


Given integers $n,a$ such that the Jacobi symbol $\left(\frac an\right)=1$, output either a square root of $a$ modulo $n$, or a nontrivial factor of $n$.
Any PPA-complete problem, such as: given a circuit $C\colon\{0,1\}^m\to\{0,1\}^m$ computing an involution (i.e., $C(C(x))=x$ for every $x\in\{0,1\}^m$) such that $C(1^m)=1^m$, output an $x\ne1^m$ such that $C(x)=x$.
The weak pigeonhole principle: given a circuit $C\colon\{0,1\}^{2m}\to\{0,1\}^m$, output $x\ne y\in\{0,1\}^{2m}$ such that $C(x)=C(y)$.