a. There are 9039 nude numbers below 1,000,000: the 9030 counted by the program below, plus the nine trivial cases of single-digit numbers 1 through 9, making 9039.  Although the program went through 10,000,000, only those numbers below 1,000,000 were counted for purposes of part a. In case you're interested a modified version counts 52,981 under 10,000,000 other than the trivial 9.

b. The smallest nude number containing all the odd digits is 1117935. Other such numbers are shown in the output below, all being seven digits in length, the largest tested.

c. The smallest triplet of consecutive nontrivial nude numbers is 1111, 1112, 1113. After that the next two such triplets have seven digits in each of their members; they are listed below as part of the output.

d. All repunit numbers are nude as are all those of the form 11...112 or 11...115, and those with a number of 1's that's a multiple of three, followed by a 3 or a 6 or a multiple of 9 1's followed by a 9 plus such numbers with groups of three 1's replaced by a 3, and similar such.

Three in a row:

 1111 1112 1113

 1111111 1111112 1111113

 1111115 1111116 1111117

 

Having all the odd digits: 

1117935

1131795

1197315

1311975

1371195

1711395

1713915

1739115

1911735

7119315

7131915

7191135

7913115

9111375

9173115

9311715

9711135

Count of those over 10 and under 1,000,000:

9030 

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Form1.Visible = True

 Text1.Text = ""

 crlf = Chr(13) + Chr(10)

 

 For n = 10 To 10000000

    DoEvents

    s$ = LTrim(Str(n))

    If InStr(s, "0") = 0 Then

       good = 1

       For i = 1 To Len(s)

         dvr = Val(Mid(s, i, 1))

         If n / dvr > Int(n / dvr) Then good = 0: Exit For

       Next

       If good Then

         If n < 1000000 Then nudeCt = nudeCt + 1

         inRow = inRow + 1

         If inRow >= 3 Then

           For j = n - inRow + 1 To n

             Text1.Text = Text1.Text & Str(j)

           Next

           Text1.Text = Text1.Text & crlf

         End If

         hasAll = 1

         For i = 1 To 5

           If InStr(s, Mid("13579", i, 1)) = 0 Then hasAll = 0: Exit For

         Next

         If hasAll Then Text1.Text = Text1.Text & n & crlf

       Else

         inRow = 0

       End If

    Else

      inRow = 0

    End If

 Next

 

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

  

End Sub