Continued Fraction of Copeland-Erdos

I was alerted to the existence of the Copeland-Erdos constant by way of my work with the Champernowne constant. Three of Sloane's associated integer sequences had a paucity of terms that I felt I might be able to add to (by using Mathematica 4).

ce = ContinuedFraction[ToExpression[StringJoin[".", Table[ToString[Prime[i]], {i, 1, 500000}]]]];

In about 12 hours (on a 300 MHz G3), this gives us 3307255 terms of the continued fraction (based on the first 500000 primes). The bulk of the calculation (the string-joining/to-expression) is pretty intense, taking total control of the CPU's clock cycles. An arithmetical re-write of the program allows it to run in the background, but takes twice as long. I should be able to do one million primes (in the next few weeks), requiring (methinks) a 48-hour window where I will not be using my computer.

Complement[Range[1500], ce]

{1026, 1128, 1135, 1145, 1152, 1165, 1257, 1298, 1307, 1332, 1356, 1363, 1367, 1396, 1404, 1425, 1432, 1440, 1443, 1445, 1446, 1454, 1471, 1475, 1496}

...the smallest integers that have yet to appear.

Table[First[Flatten[Position[ce, i]]], {i, 1, 1025}]

{8, 16, 20, 2, 7, 15, 12, 4, 17, 254, 98, 154, 53, 212, 253, 5, 83, 6, 64, 353, 1049, 555, 562, 690, 88, 334, 54, 1795, 621, 392, 1101, 1203, 715, 3929, 628, 138, 257, 648, 29, 329, 59, 2417, 444, 1353, 342, 1348, 911, 3043, 3870, 840, 990, 681, 229, 689, 749, 2311, 8254, 18, 1976, 2460, 395, 233, 791, 496, 998, 3479, 2118, 157, 1534, 3439, 7070, 274, 2703, 1519, 3779, 730, 694, 4654, 1562, 262, 295, 718, 4633, 1824, 1051, 1229, 36, 5129, 2556, 939, 4865, 6313, 4493, 14910, 11038, 16317, 5155, 3769, 5273, 6681, 1156, 227, 4331, 6558, 5555, 2170, 4872, 8529, 26082, 11106, 8806, 17088, 18558, 9310, 12499, 4377, 13139, 2095, 10324, 31143, 876, 8409, 4371, 25053, 12434, 7073, 15961, 17437, 3376, 2486, 42871, 4931, 4746, 13487, 1068, 29332, 34811, 2403, 1693, 13040, 15567, 3305, 203, 21736, 2688, 26797, 138804, 9657, 17079, 6567, 6796, 726, 15522, 24757, 8186, 5029, 1239, 18828, 3238, 1682, 9515, 24834, 23161, 4687, 168, 4672, 6651, 4763, 3545, 446, 9531, 11741, 6836, 18137, 22353, 13175, 14300, 28640, 26321, 19969, 38632, 14697, 23329, 28719, 15995, 17853, 524, 38447, 19191, 26425, 134, 11279, 17127, 6811, 43283, 38818, 16102, 102556, 2860, 1355, 39611, 15909, 32918, 12548, 3294, 10862, 21758, 16736, 64125, 3573, 28623, 17817, 70535, 18502, 10713, 22459, 4448, 94875, 85062, 9442, 15920, 86196, 7761, 7912, 2730, 31736, 103653, 999, 66319, 16186, 1131, 138595, 29513, 138250, 7989, 94841, 52223, 84205, 182343, 19919, 30866, 20963, 3804, 62613, 30196, 259, 42512, 132428, 90787, 7038, 54124, 15935, 29243, 28953, 1967, 154406, 21093, 2374, 36081, 75240, 37792, 107025, 37921, 32437, 1446, 85785, 71811, 12215, 96375, 4864, 51734, 1661, 36156, 4160, 29133, 10899, 18728, 23459, 5483, 30703, 27787, 106577, 61093, 79055, 89524, 1849, 43095, 103936, 5804, 57354, 98972, 40956, 118935, 21005, 1456, 30297, 25275, 21569, 29946, 2503, 113579, 34247, 18060, 23612, 34333, 18246, 200712, 40537, 1690, 212358, 21502, 171057, 108493, 49299, 184219, 26059, 7382, 202128, 98892, 26273, 193137, 29468, 30944, 23866, 192057, 51294, 40095, 114485, 85803, 77365, 242205, 106174, 39774, 24999, 27070, 73762, 49004, 103001, 110904, 32655, 17195, 30895, 54179, 62858, 48542, 33222, 61939, 51455, 54567, 194948, 30571, 103864, 29261, 93170, 5317, 21693, 25270, 136866, 77293, 7300, 33664, 94189, 93110, 14257, 86451, 87540, 24526, 219434, 21577, 247895, 3056, 23809, 47158, 1777, 147464, 103795, 557725, 7848, 224830, 223701, 266897, 327347, 16508, 21606, 135692, 64537, 19084, 222256, 102066, 55660, 49059, 348286, 265, 23393, 4846, 169023, 303824, 20371, 88745, 25692, 113025, 216635, 48526, 203111, 221945, 23969, 9566, 75390, 68630, 145123, 27005, 121600, 145540, 393407, 155098, 61514, 237731, 67900, 12331, 2753, 129382, 790035, 3662, 41946, 128682, 8191, 318700, 12368, 233682, 20569, 43517, 39682, 211172, 184205, 82031, 151355, 107061, 191735, 254886, 40074, 108929, 81983, 308559, 134962, 127593, 118930, 12812, 333424, 184927, 122174, 289265, 29440, 31807, 169014, 80750, 30161, 13907, 39474, 15609, 245451, 88877, 62969, 104757, 279336, 54716, 112566, 80746, 740524, 44974, 99297, 12420, 47419, 72483, 149163, 181421, 49968, 118660, 192946, 183894, 367371, 207212, 96341, 220955, 72, 51728, 364993, 209465, 94807, 22641, 28360, 126460, 15665, 657600, 34585, 12279, 146612, 431533, 319801, 41008, 307049, 188565, 407345, 341865, 157429, 147750, 190932, 455683, 10644, 185495, 20938, 320141, 19111, 98294, 195707, 100961, 17862, 139178, 36090, 178917, 255070, 120356, 68180, 66127, 141932, 213146, 86688, 227874, 486185, 20424, 76726, 42060, 35316, 37890, 398534, 187137, 348617, 285603, 58366, 415642, 67186, 59250, 538048, 490473, 10781, 103513, 55336, 142441, 491300, 187752, 510711, 71393, 65808, 117389, 170184, 138641, 430769, 95697, 87044, 227047, 786567, 258428, 51287, 210589, 130390, 248429, 156935, 148010, 99024, 7810, 503152, 631585, 93279, 210282, 4528, 272236, 19694, 395946, 124596, 44841, 82030, 139465, 141684, 137914, 307448, 94952, 528184, 154660, 24929, 870229, 272713, 34689, 250718, 34068, 124186, 326667, 605777, 86247, 128801, 221098, 713769, 1185555, 281542, 75966, 237385, 264690, 609103, 22081, 1829, 102318, 376494, 438016, 180136, 354475, 234123, 69886, 8859, 121052, 29702, 72262, 165858, 129959, 130492, 111806, 498825, 7104, 154086, 59714, 802828, 468584, 80066, 222623, 32266, 90499, 52316, 180382, 159749, 613556, 213954, 628372, 409884, 152747, 741191, 611931, 579712, 834892, 122723, 159472, 780897, 332488, 581101, 302609, 674072, 140299, 172361, 105391, 969107, 3299, 281221, 71831, 903372, 216459, 394518, 98370, 348133, 98359, 214304, 782844, 1448923, 582240, 156310, 74128, 93699, 43692, 516380, 739151, 25435, 80265, 139865, 768716, 602358, 87615, 79872, 287947, 133375, 700339, 68960, 101831, 524001, 338001, 552094, 577098, 29164, 414203, 320109, 263471, 1233586, 67795, 443765, 360263, 210593, 2452, 311380, 47165, 52365, 567662, 219565, 795318, 253993, 40037, 262934, 99923, 475201, 80594, 389544, 145944, 269095, 511708, 200640, 341096, 244794, 274438, 81069, 1489641, 136012, 353101, 77215, 887533, 95511, 455273, 87365, 29956, 1000309, 140796, 235248, 117452, 1681229, 377155, 715831, 721116, 463793, 33409, 454346, 857798, 8868, 81683, 154644, 180296, 19278, 133045, 322040, 288221, 50496, 301241, 249703, 514869, 173846, 377683, 501721, 263168, 107579, 398799, 899894, 419962, 266178, 967404, 16138, 294168, 65533, 654445, 114365, 344666, 1445764, 494258, 83912, 167411, 1808072, 192238, 122390, 180170, 1144031, 16505, 212698, 743507, 165595, 704591, 144981, 833752, 1026140, 17443, 1343691, 1118289, 673556, 947302, 42352, 677840, 240995, 54911, 1984862, 305246, 742584, 964926, 159953, 156590, 231135, 610543, 494887, 150849, 17435, 1351499, 52958, 267138, 211446, 447844, 48163, 25983, 195828, 90273, 100430, 181767, 404394, 406513, 31087, 90405, 1505445, 757514, 556520, 500466, 762142, 1007792, 71315, 182184, 10419, 36774, 504693, 812944, 811769, 198467, 976565, 959038, 1654418, 844727, 317306, 58192, 359483, 1512154, 107677, 640005, 177758, 18966, 2246283, 111357, 228243, 251323, 419738, 884100, 190850, 3936, 356398, 84581, 461749, 558444, 31740, 210214, 21915, 399874, 32542, 528094, 245865, 2017132, 179011, 623878, 100737, 44439, 1488394, 384654, 878150, 1542929, 1238819, 211104, 196952, 336766, 746019, 192016, 764726, 370724, 118972, 1598087, 1722898, 118917, 2005607, 172201, 902160, 104012, 196935, 666497, 40123, 2586784, 20009, 86590, 5549, 670805, 654918, 817179, 814881, 743776, 881833, 1781847, 15014, 1079243, 100959, 288980, 95097, 457877, 74087, 43603, 43311, 2033881, 212339, 527831, 299186, 1316295, 63420, 44483, 166244, 557167, 411863, 394258, 2446440, 161339, 7970, 1537015, 406594, 1057432, 545652, 6620, 352370, 46354, 166973, 61690, 54206, 310634, 747906, 114804, 641586, 195422, 607353, 652597, 188089, 50133, 157577, 830358, 1093861, 307625, 1351390, 1508156, 51625, 390257, 202941, 326570, 862858, 1772810, 655656, 93043, 12329, 805056, 852178, 264134, 604873, 1184870, 701739, 151467, 1086578, 248661, 793731, 44303, 174689, 470340, 483214, 123216, 69620, 116960, 1141821, 63786, 47567, 240402, 2157328, 2074713, 803715, 939285, 841783, 222340, 2234761, 510166, 796582, 228458, 7471, 659623, 2047, 292361, 139234, 239184, 780734, 191878, 19106, 12229, 408129, 181921, 721912, 774555, 338479, 1265035, 534144, 706538, 433618, 524459, 399484, 115525, 178977, 123495, 621508, 501177, 547313, 722219, 629669, 187776}

...the positions (= index+1) of first occurrence of the first 1025 integers in the Copeland-Erdos Continued-Fraction expansion. [A033309]

j = 0; t = Table[If[ce[[i]] > j, i; j = ce[[i]]], {i, 1, Length[ce]}]; v = Select[t, IntegerQ]

{4, 8, 16, 18, 58, 87, 484, 1468, 3955, 6986, 17474, 26174, 37084, 73646, 89141, 187645, 542374, 551071, 644045, 925191, 2037817, 4576230}

...the incrementally-largest terms in ce. [A033310]

Table[First[Flatten[Position[ce, v[[i]]]]], {i, 1, Length[v]}]

{2, 4, 5, 6, 18, 36, 72, 89, 1557, 3728, 4730, 27642, 60322, 90720, 104378, 107183, 241453, 453796, 679114, 901262, 934243, 1298092}

...the positions (= index+1) in ce where these incrementally-largest terms occur. [A033311]