The program (at the bottom) was designed to use only up to the 50th prime. That means that in order to insure completeness for all reported numbers, those numbers should not exceed 456, as the sum of the 49th and 50th primes.  Also, as a further consequence, only a running count for up to 18 primes would be necessary, as 19 or more consecutive primes would add up to more than 456.

The intended answer is probably 240, as:

240 = 17+19+23+29+31+37+41+43

= 53+59+61+67

= 113+127

This is reported as the entry below:

240

7 14:   17 19 23 29 31 37 41 43

16 19:   53 59 61 67

30 31:   113 127

where the numbers before the colon refer to the ordinality of the primes that follow; for example 17 is the 7th prime and 43 is the 14th.

287 would be the next such number.

In the entries below, those in which the target number is also prime are marked with **. That's to indicate that the number itself could be considered a degenerate "sum" of consecutive primes. If that were done, 41 would be the lowest, as there are two other sums of consecuteve primes that add up to 41.

Interestingly 311 has four sums of actual consecutive primes that add up to it, plus it is itself prime, giving a 5th, degenerate, sum.

41 **

1 6:   2 3 5 7 11 13

5 7:   11 13 17

83 **

5 9:   11 13 17 19 23

9 11:   23 29 31

197 **

1 12:   2 3 5 7 11 13 17 19 23 29 31 37

7 13:   17 19 23 29 31 37 41

199 **

11 15:   31 37 41 43 47

18 20:   61 67 71

223 **

8 14:   19 23 29 31 37 41 43

20 22:   71 73 79

240

7 14:   17 19 23 29 31 37 41 43

16 19:   53 59 61 67

30 31:   113 127

251 **

9 15:   23 29 31 37 41 43 47

22 24:   79 83 89

281 **

1 14:   2 3 5 7 11 13 17 19 23 29 31 37 41 43

10 16:   29 31 37 41 43 47 53

287

7 15:   17 19 23 29 31 37 41 43 47

15 19:   47 53 59 61 67

24 26:   89 97 101

311 **

5 15:   11 13 17 19 23 29 31 37 41 43 47

11 17:   31 37 41 43 47 53 59

16 20:   53 59 61 67 71

26 28:   101 103 107

340

7 16:   17 19 23 29 31 37 41 43 47 53

10 17:   29 31 37 41 43 47 53 59

39 40:   167 173

371

4 16:   7 11 13 17 19 23 29 31 37 41 43 47 53

13 19:   41 43 47 53 59 61 67

30 32:   113 127 131

401 **

10 18:   29 31 37 41 43 47 53 59 61

14 20:   43 47 53 59 61 67 71

439 **

11 19:   31 37 41 43 47 53 59 61 67

34 36:   139 149 151

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Text1.Text = ""

 crlf$ = Chr(13) + Chr(10)

 Form1.Visible = True

 hiPrime = 50

 maxGoal = prm(hiPrime - 1) + prm(hiPrime)

 maxCt = 0

 While tot <= maxGoal

   maxCt = maxCt + 1

   tot = tot + prm(maxCt)

 Wend

 Text1.Text = Text1.Text & maxCt & Str(maxGoal) & crlf

 ReDim totPrm(maxCt), ways(maxGoal, 10, 2)

 For p = 1 To hiPrime

   For i = 2 To maxCt

     totPrm(i) = totPrm(i) + prm(p)

     prevP = p - i

     If prevP > 0 Then

      totPrm(i) = totPrm(i) - prm(prevP)

     End If

     If p >= i Then

       goal = totPrm(i)

       If goal <= maxGoal Then

         ways(goal, 0, 0) = ways(goal, 0, 0) + 1

         s = ways(goal, 0, 0)

         ways(goal, s, 1) = prevP + 1

         ways(goal, s, 2) = p

       End If

     End If

   Next

 Next p

 

 For i = 1 To maxGoal

   If ways(i, 0, 0) > 2 Or prmdiv(i) = i And ways(i, 0, 0) > 1 Then

    Text1.Text = Text1.Text & i

    If prmdiv(i) = i Then Text1.Text = Text1.Text & " **"

    Text1.Text = Text1.Text & crlf

    For j = 1 To ways(i, 0, 0)

     Text1.Text = Text1.Text & ways(i, j, 1) & Str(ways(i, j, 2)) & ":  "

     For k = ways(i, j, 1) To ways(i, j, 2)

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

     Next

     Text1.Text = Text1.Text & crlf

    Next

    Text1.Text = Text1.Text & crlf

   End If

 Next

 

End Sub

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