# [R], <s>85</s> 81 bytes

<!-- language-all: lang-r -->

    while(T<-T+1)grepl("^((.+)\\2+)+$",Reduce(paste0,T%/%2^(0:log2(T))%%2))&&print(T)

[Try it online!][TIO-ks7vsg1t]

[R]: https://www.r-project.org/
[TIO-ks7vsg1t]: https://tio.run/##DcdBCoAgEADAv0jKLlqZx@gV4TGEqMUEKTGj51tzm1zre4RIYKfWygF9phSBOYBO4rIYibJhaqb92QjSehfSyvKeGwd6jJc3YBE5N4hCpBzO8r/WDw "R – Try It Online"

A regex-based solution. Prints values indefinitely.

Thanks to @pajonk for saving 4 bytes!