Special case 1: b = p^e, e>1 -> x = p and k/(b-1) = p^(e-1)
b = p^e appears to be the largest of a family of bases whose k/(b-1) = p^(e-1). The others are greater than p^(e-1) and divisible by p. I've labelled four of these families in my graph: k/(b-1) = 3^6: b = {732, 735, 738, 741, 744, 747, 750, 753, 756, 759, 762, 765, 768, 771, 777, 783, 789, 795, 801, 807, 813, 819, 825, 831, 837, 843, 849, 855, 861, 867, 873, 879, 885, 891, 897, 903, 909, 915, 921, 927, 933, 939, 951, 963, 981, 987, 993, 999, 1005, 1011, 1017, ...}, presumably not complete until we get to base 2187 = 3^7. k/(b-1) = 5^4: b = {630, 635, 640, 645, 655, 665, 675, 685, 695, 715, 725, 745, 755, 775, 785, 805, 815, 835, 845, 865, 875, 895, 905, 925, 935, 955, 965, 985, 995, 1025, ...}, presumably not complete until we get to base 3125 = 5^5. k/(b-1) = 2^9: b = {514, 516, 518, 520, 522, 524, 526, 528, 530, 532, 534, 536, 538, 540, 542, 544, 546, 548, 550, 554, 556, 558, 560, 562, 564, 566, 568, 572, 574, 576, 580, 584, 586, 590, 592, 596, 602, 604, 608, 610, 614, 620, 622, 626, 628, 632, 634, 638, 652, 656, 658, 662, 664, 668, 674, 676, 682, 686, 688, 692, 694, 698, 704, 706, 712, 716, 718, 724, 734, 746, 752, 758, 764, 766, 772, 776, 778, 788, 794, 796, 802, 808, 818, 824, 838, 842, 844, 848, 856, 862, 866, 872, 878, 886, 892, 898, 904, 908, 914, 916, 922, 926, 932, 934, 944, 956, 958, 964, 974, 976, 982, 998, 1004, 1006, 1016, 1018, 1024}, presumably complete because 1024 = 2^10. k/(b-1) = 7^3: b = {350, 357, 364, 371, 378, 385, 392, 413, 427, 441, 455, 469, 497, 511, 539, 553, 581, 623, 637, 679, 707, 721, 749, 763, 791, 833, 889, 917, 959, 973, ...}, presumably not complete until we get to base 2401 = 7^4.