Try adding three lines of code to put the formula back like:
Option Explicit
Sub ArchiveRecords()
Dim RowCtr As Double
Dim LastRowI As Double
Dim LastRowA As Double
LastRowI = Cells(Rows.Count, "A").End(xlUp).Row
For RowCtr = 4 To LastRowI
If Cells(RowCtr, "E") <> "" Then
LastRowA = Worksheets("Archive").Cells(Rows.Count, "A").End(xlUp).Row + 1
Range(Cells(RowCtr, "A"), Cells(RowCtr, "E")).Copy _
Destination:=Worksheets("Archive").Cells(LastRowA, "A")
Rows(RowCtr).EntireRow.Delete
RowCtr = RowCtr - 1
End If
Next RowCtr
LastRowI = Cells(Rows.Count, "A").End(xlUp).Row
Cells(5, "F").Formula = "=F4+1" 'Formula of your choice
Cells(5, "F").Copy Destination:=Range("F6:F" & LastRowI)
End Sub
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