The program finds

1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/21 + 1/135 + 1/10395

1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/21 + 1/165 + 1/693

1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/21 + 1/231 + 1/315

1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/33 + 1/45 + 1/385

1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/35 + 1/45 + 1/231

The whole output was

2.828968253968251

    + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/21 + 1/135 + 1/10395

1

    + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/21 + 1/165 + 1/693

1

    + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/21 + 1/231 + 1/315

1

    + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/33 + 1/45 + 1/385

1

    + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/15 + 1/35 + 1/45 + 1/231

19999

 done

 

 

The first number printed was just a verification that the reference table was constructed correctly, by checking its highest entry. The table was used in the program to see if there were sufficiently large numbers in the next few to get to at least 1 as a total. Combined with stopping the recursion if the total got to over 1 or the number of terms would get to more than 9, this speeded up processing.

The number 19999 at the end indicated we reached the limit of the entries we were trying for. If it had stopped short of that value, we could say we reached a natural limit and that there were no further answers. But that's not the case so we haven't show that we have all the answers. This was probably to be expected as the series diverges, so this method would never show a lack of possibility for further solutions.

Discussions while in the queue, with reference to outside web sites, indicate these are the only five possibilities, but this program does not prove that. The very determination to go to 20,000 was based on a knowledge of the answer, and so is really a cheat.

DefDbl A-Z

Dim crlf$, h(36), tot, sumNext(1 To 9, 20000), highI

Private Sub Form_Load()

 Form1.Visible = True

 

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 

 For i = 20000 To 1 Step -1

   sumNext(1, i) = 1 / i

   For n = 2 To 9

     For j = i To i + n - 1

       If j <= 20000 Then

         sumNext(n, i) = sumNext(n, i) + sumNext(1, j)

       End If

     Next

   Next

 Next

 

 Text1.Text = Text1.Text & sumNext(9, 1)

 For st = 1 To 101 Step 2

   h(1) = st: tot = 1 / st

   addOn 2

 Next

 

 Text1.Text = Text1.Text & crlf & highI & crlf

 

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

  

End Sub

Sub addOn(wh)

  DoEvents

  st = h(wh - 1) + 2

  For i = st To 20000 Step 2

    DoEvents

    h(wh) = i

    savetot = tot

    tot = tot + sumNext(1, i)

    If Abs(tot - 1) < 0.0000000000001 And wh = 9 Then

       Text1.Text = Text1.Text & tot & crlf & "   "

       For j = 1 To wh

         Text1.Text = Text1.Text & " + 1/" & h(j)

       Next

       Text1.Text = Text1.Text & crlf

    End If

    If i < 20000 And tot < 1.0001 And wh < 9 Then

     If tot + sumNext(9 - wh, i + 1) + 0.000000001 >= 1 Then

      If i > highI Then highI = i

      addOn wh + 1

     End If

    End If

    tot = savetot

  Next

End Sub