Let S(k) = 2 + 3 + 5 + 7 + 11 + 13 ... + p(k) , p(k) being the k-th prime.
For what values of k is S(k) a palindrome?

Please ignore the trivial answers k =1 and k=2.
Go as high as you can.

  10        repeat
  20          P=nxtprm(P)
  30          inc K
  40          S=S+P
  50          Sstr=cutspc(str(S))
  60          Good=1
  70          for I=1 to int(len(Sstr)/2)
  80             if mid(Sstr,I,1)<>mid(Sstr,len(Sstr)+1-I,1) then Good=0
  90          next
 100          if Good then print K;P;S
 110        until P=0
 
finds before being stopped manually (each line shows k, p(k), S(k)):

 1  2  2
 2  3  5
 8  19  77
 7693  78347  285080582
 8510  87641  352888253
 12941  139241  854848458
 146134  1959253  137372273731
 637571  9564097  2939156519392

which is enough to find:

From Sloane's OEIS (A038582, A038583, A038584)

       k               p(k)                    S(k)
           1                 2                         2  
           2                 3                         5  
           8                19                        77  
        7693             78347                 285080582  
        8510             87641                 352888253  
       12941            139241                 854848458  
      146134           1959253              137372273731  
      637571           9564097             2939156519392  
    27198825         516916921          6833383883833386  
    53205635        1048924213         27155268786255172  
  6283318531      155353386241     477749724515427947774  
  7167375533      178196630873     625179415050514971526  

It would have taken quite a while for my computer to have reached this high.

Posted by Charlie on 2012-04-21 14:12:08