This program

   10   repeat

   20     P=nxtprm(P)

   30     C=16*P+1

   40     Cr=int(C^(1/3)+0.5)

   50     if Cr*Cr*Cr=C then print P,C,Cr

   60   until 0

found 16*307 + 1 = 4913 = 17^3 almost right away. It's now chugging along without having found any more. I'm about to check the current value of P: it's 583969 and still going without finding a second one.