Is it true that any smooth manifold of dimension $n$ can be smoothly embedded into $\mathbb{R}^{2n+1-a(n)}$ where $a(n)$ is the number of appearances of digit "1" in the binary expansion of $n$?



Making immersions from immersion conjecture into embeddings
The following result was proved by Ralph Cohen [1]:



> 
> **Theorem.** Let $\alpha(n)$ be the number of ones in the binary representation of $n$, and $M^n$ is a compact $C^\infty$,
>  $n$-manifold, $n>1$. Then there is a smooth immersion of $M^n$ to
>  $\mathbb{R}^{2n-\alpha(n)}$.
> 
> 
> 


Whether, there is an embedding into $\mathbb{R}^{2n-\alpha(n)+1}$, I do not know.


[1] Cohen, Ralph L.
The immersion conjecture for differentiable manifolds. 
Ann. of Math. (2) 122 (1985), no. 2, 237–328. 


