The denominators are the squares of:

as found by the below VB program and verified by using exact rational fractions in UBASIC.

The program  produces a total that prints out as exactly .5 but did not register as equal to .5 within the program, due to rounding, and thus necessitated the difference to be below the small threshhold of absolute value difference shown in the program, rather than testing if = 0.5.

The first portion of the program determined how far back from 1/36^2 would be necessary and found that the first term must be 1/2^2, otherwise the total would be impossible. It was only after this was run that the rest of the program was added, to find the actual result.

DefDbl A-Z

Dim crlf$, h(36), tot

Private Sub Form_Load()

 Form1.Visible = True

 

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 

 For rt = 36 To 1 Step -1

   ptot = tot

   tot = tot + 1 / (rt * rt)

   If tot >= 0.5 Then Text1.Text = Text1.Text & rt & Str(tot) & Str(ptot) & crlf & crlf: Exit For

 Next

 

 h(1) = 4: tot = 1 / 4

 addOn 2

 

 

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

  

End Sub

Sub addOn(wh)

  DoEvents

  st = Int(Sqr(h(wh - 1)) + 0.5) + 1

  For i = st To 36

    h(wh) = i * i

    savetot = tot

    tot = tot + 1 / (i * i)

    If Abs(tot - 0.5) < 0.000000000001 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

       For j = 1 To wh

         Text1.Text = Text1.Text & Str(Sqr(h(j)))

       Next

       Text1.Text = Text1.Text & crlf

       

    End If

    If i < 36 And tot < 0.5001 Then

      addOn wh + 1

    End If

    tot = savetot

  Next

End Sub

2 .617538519845469 .367538519845469

0.5

    + 1/4 + 1/9 + 1/16 + 1/25 + 1/49 + 1/144 + 1/225 + 1/400 + 1/784 + 1/1225

 2 3 4 5 7 12 15 20 28 35