Here are the 23 integers from 1 through 100 that are represented in base 3 by only the digits 0 and 1:

 1      1

 3     10

 4     11

 9    100

10    101

12    110

13    111

27   1000

28   1001

30   1010

31   1011

36   1100

37   1101

39   1110

40   1111

81  10000

82  10001

84  10010

85  10011

90  10100

91  10101

93  10110

94  10111

These are the only positive integers less than or equal to 100 that can be written as the sum of distinct powers of 3, so there are 100 - 23 = 77 that cannot.

Note that if we were doing this analytically we would start by representing 100 as base-3: that's 10201. The highest base-3 number under this that uses only 0's and 1's is 10111. From there we can treat it as binary, as all binary numbers less than this are valid. 10111 is binary for 23, the number of such configurations, whether interpreted as binary or as ternary--the same 23 we counted.

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Form1.Visible = True

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 For n = 1 To 100

   r$ = LTrim(base$(n, 3))

   good = 1

   For i = 1 To Len(r)

     If InStr("01", Mid(r, i, 1)) = 0 Then good = 0: Exit For

   Next

   If good Then

     ct = ct + 1

     Text1.Text = Text1.Text & n & "  " & r & crlf

   End If

 Next

 

 Text1.Text = Text1.Text & ct & crlf & " done"

  

End Sub

Function base$(n, b)

  v$ = ""

  n2 = n

  Do

    d = n2 Mod b

    n2 = n2 \ b

    v$ = Mid("0123456789abcdefghijklmnopqrstuvwxyz", d + 1, 1) + v$

  Loop Until n2 = 0

  base$ = v$

End Function