> What is the length $f(n)$ of the shortest nontrivial group word $w_n$ in $x_1,\ldots,x_n$ that collapses to $1$ when we substitute $x_i=1$ for any $i$?

For example, $f(2)=4$, with the commutator $[x_1,x_2]=x_1 x_2 x_1^{-1} x_2^{-1}$ attaining the bound.  

For any $m,n \ge 1$, the construction $w_{m+n}(\vec{x},\vec{y}):=[w_m(\vec{x}),w_n(\vec{y})]$ shows that $f(m+n) \le 2 f(m) + 2 f(n)$.

Is $f(1),f(2),\ldots$ the same as sequence [A073121][1]:
$$ 1,4,10,16,28,40,52,64,88,112,136,\ldots ?$$

**Motivation:** Beating the iterated commutator construction would improve the best known bounds in [size of the smallest group not satisfying an identity][2].


  [1]: http://oeis.org/A073121 "A073121"
  [2]: http://mathoverflow.net/questions/15022/size-of-the-smallest-group-not-satisfying-an-identity/15065#15065