The smallest is 104869. Listed below are the first 100 meeting the criterion in the text of the puzzle. Those also satisfying the title of the puzzle are marked with an asterisk. 1043869 is the first one that doesn't fit the title as well, as it includes a digit 3.

104869 *

108649 *

140689 *

140869 *

148609 *

164089 *

164809 *

168409 *

184609 *

186049 *

401689 *

406981 *

408169 *

408691 *

409861 *

416089 *

418069 *

460189 *

460891 *

460981 *

468019 *

468109 *

469801 *

480169 *

486091 *

489061 *

498061 *

601849 *

604189 *

604819 *

608941 *

610849 *

618049 *

640891 *

641089 *

648019 *

649081 *

649801 *

681049 *

681409 *

684091 *

684109 *

689041 *

690841 *

694081 *

801469 *

804619 *

806941 *

809461 *

814069 *

814609 *

841069 *

849061 *

849601 *

860941 *

864091 *

864901 *

904681 *

904861 *

906481 *

946081 *

946801 *

948061 *

964081 *

968041 *

980641 *

1014869 *

1016489 *

1016849 *

1018649 *

1041869 *

1043869

1044689 *

1046189 *

1046389

1046849 *

1046897

1047689

1048609 *

1048963

1049681 *

1049683

1049687

1049861 *

1049863

1061849 *

1063849

1064689 *

1064989 *

1067489

1067849

1068149 *

1068409 *

1068439

1068469 *

1068491 *

1068497

1068499 *

1068941 *

1080649 *

...

p = 1

 Do

   p = nxtprm(p)

   s$ = LTrim(Str(p))

   good1 = 1: good2 = 1

   For i = 1 To 6

     If InStr(s, Mid("014689", i, 1)) = 0 Then good1 = 0

   Next

   For i = 1 To 4

     If InStr(s, Mid("2357", i, 1)) > 0 Then good2 = 0

   Next

   

   If good1 Then

     Text1.Text = Text1.Text & p

     If good2 Then Text1.Text = Text1.Text & " *"

     Text1.Text = Text1.Text & crlf

     ct = ct + 1

   End If

   DoEvents

 Loop Until ct = 100

...