TI-Basic, 45 bytes

Input NFor(B,2,NIf prod(seq(BfPart(iPart(N/B^X)/B),X,0,log(N)/log(B))<2Disp BEnd

Explanation

• Input N
• For every B from 2 to N
  • If N is just 0 and 1 in base B
    • Display B
• End loop

The complicated part

The second line works as follows:

• For every X from 0 to log_B N
• B × fPart(iPart(N / B^X) / B) is the Nth digit in base B, counting backwards
• Consider this as a list
• For each element, if the digit is less than 2, yield 1 (true), else 0 (false)
• Take the product: 1 iff all elements are 1

Note

The program runs significantly faster if a closing parenthesis ) is placed at the end of the second line. See here for more about this.