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This paper is devoted to establishing the compactness and existence resultsof the solutions to the prescribing fractional $Q$-curvatures problem of order$2\sigma$ on $n$-dimensional standard sphere when $ n-2\sigma=2$,$\sigma=1+m/2,$ $m\in \mathbb{N}_{+}.$ The compactness results are novel andoptimal. In addition, we prove a degree-counting formula of all solutions toachieve the existence. From our results, we can know where blow up occur.Furthermore, the sequence of solutions that blow up precisely at any finitedistinct location can be constructed. It is worth noting that our resultsinclude the case of multiple harmonic.

Compactness and existence results of the prescribing fractional $Q$-curvatures problem on $\mathbb{S}^n$