You're not clear on what you're wanting the result to be,
88 89 90 91 92 93 is not the same as 88;89;90;91;92;93
but you can make it whatever you want in a macro.
This macro will fire when you type a new value into A1 or B1.
You've given no ground rules for the numbers in A & B column so I just put some in that may or may not be practical for your purpose.
Right click the sheet tab, select View Code and paste this into the sheet module.
Code:
Private Sub Worksheet_Change(ByVal Target As Range)
Dim i As Integer, str As String
'limit to single cell
If Target.Count > 1 Then Exit Sub
'limit to row 1
If Target.Row <> 1 Then Exit Sub
'limit to columns A and B
If Target.Column > 2 Then Exit Sub
'clear anything existing in C
'disable events so this macro doesn't call itself when it changes a cell
Application.EnableEvents = False
Cells(Target.Row, "C").ClearContents
're-enable events
Application.EnableEvents = True
'make sure both cells have numbers
If Not IsNumeric(Cells(Target.Row, "A")) Or Not IsNumeric(Cells(Target.Row, "B")) Then Exit Sub
'make sure A cell is smaller than B cell
If Cells(Target.Row, "A") >= Cells(Target.Row, "B") Then Exit Sub
'put together what's to go into C1
For i = Cells(Target.Row, "A").Value To Cells(Target.Row, "B").Value
str = str & ";" & i
Next i
Application.EnableEvents = False
'write the string to C, remove the leading ;
Cells(Target.Row, "C").Value = Mid(str, 2)
Application.EnableEvents = True
End Sub
Let me know if this is something you can work with.