A binary De Bruijn sequence of index $n$ is a circular sequence $S=a_1a_2\ldots a_{2^n}$, with $a_i∈\{0,1\}$, and such that each of the $2^n$ binary $n$-tuples occurs exactly once in $S$.
What is the slowest growing sequence $n_1$, $n_2$, $\ldots$ such that there exist the sequence $b_1b_2\ldots$ where $b_1b_2\ldots b_{2^{n_i}}$ is a binary De Bruijn sequence for each $n_i$?
This question already arose in the comments to Nested De Bruijn Sequences. It is interesting because it can be applied to construction of normal numbers with small discrepancy. In the article "Concerning some questions of uniform distribution" Korobov already used De Bruijn sequences for this purpose but he wrote these sequences one after another.
