The program below checks all such sets of consecutive primes that begin at or below the 4000th prime and have 200 or fewer members. Other than 3+5 = 8 (the trivial case) it finds only 439 + 443 + 449 = 1331 = 11^3.

 3 5    8

 439 443 449    1331

 

 DefDbl A-Z

 Dim crlf$, total(4000, 200), palin

 Function mform$(x, t$)

   a$ = Format$(x, t$)

   If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$

   mform$ = a$

 End Function

 

 Private Sub Form_Load()

  Text1.Text = ""

  crlf$ = Chr(13) + Chr(10)

  Form1.Visible = True

  

  pid = 11260

 

  p = 1

  For pr = 1 To 4200

    p = nxtprm(p)

    For row = 1 To 4000

      st = pr - row + 1

      If st < 0 Then st = 0

      For col = st To 200

         If pr >= row Then

         total(row, col) = total(row, col) + p

         End If

      Next col

      DoEvents

    Next row

  Next pr

  Text1.Text = Text1.Text & "phase 2" & crlf

  

  For row = 1 To 4000

  For col = 2 To 200

    DoEvents

     If isPalin(total(row, col)) Then

        cr = Int(palin ^ (1 / 3) + 0.5)

        If cr * cr * cr = palin Then

           firstp = row: lastp = row + col - 1

           p = prm(row)

           For i = firstp To lastp

             Text1.Text = Text1.Text & Str(prm(i))

           Next

           Text1.Text = Text1.Text & "    " & total(row, col) & crlf

        End If

     End If

  Next

  Next

  

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

 

 End Sub

 

 Function isPalin(x)

   xs$ = LTrim(Str(x))

   good = 1

   For i = 1 To Len(xs) / 2

     If Mid(xs, i, 1) <> Mid(xs, Len(xs) + 1 - i, 1) Then good = 0: Exit For

   Next

   palin = Val(xs)

   isPalin = good

 End Function

 

 Function prm(i)

   Dim p As Long

   Open "17-bit primes.bin" For Random As #111 Len = 4

   Get #111, i, p

   prm = p

   Close 111

 End Function

 Function prmdiv(num)

  Dim n, dv, q

  If num = 1 Then prmdiv = 1: Exit Function

  n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0

  If limit <> Int(limit) Then limit = Int(limit + 1)

  dv = 2: GoSub DivideIt

  dv = 3: GoSub DivideIt

  dv = 5: GoSub DivideIt

  dv = 7

  Do Until dv > limit

    GoSub DivideIt: dv = dv + 4 '11

    GoSub DivideIt: dv = dv + 2 '13

    GoSub DivideIt: dv = dv + 4 '17

    GoSub DivideIt: dv = dv + 2 '19

    GoSub DivideIt: dv = dv + 4 '23

    GoSub DivideIt: dv = dv + 6 '29

    GoSub DivideIt: dv = dv + 2 '31

    GoSub DivideIt: dv = dv + 6 '37

  Loop

  If n > 1 Then prmdiv = n

  Exit Function

 

 DivideIt:

  Do

   q = Int(n / dv)

   If q * dv = n And n > 0 Then

     prmdiv = dv: Exit Function

    Else

     Exit Do

   End If

  Loop

 

  Return

 End Function

 Function nxtprm(x)

   Dim n

   n = x + 1

   While prmdiv(n) < n Or n < 2

     n = n + 1

   Wend

   nxtprm = n

 End Function