R, 85 bytes

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

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A regex-based solution. Prints values indefinitely. I wish R had something more concise than paste(…,collapse=“”)!

R, 85 81 bytes

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

Try it online!

A regex-based solution. Prints values indefinitely.

Thanks to @pajonk for saving 4 bytes!

R, 85 bytes

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

Try it online!

A regex-based solution. Prints values indefinitely. I wish R had something more concise than paste(…,collapse=“”)!

R, 85 bytes

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

Try it online!

A regex-based solution. Prints values indefinitely. I wish R had something more concise than paste(…,collapse=“”)!