Hello MathFun,

[hope no mistakes here, calculated by hand -- and a first warm thank you to Michael Kleber, Franklin T. Adams-Watters and Joshua Zucker]

Say we have a rule which transforms an integer in another one like this:

- Add to n its odd digits and subtract its even ones.

So 738 will become 740 (738+7+3-8)

103 will become 107 (103+1+3)

and 358 stays 358 (358+3+5-8).

Let’s loop this procedure for integers 0 to 100 and see what happens:

0 -> 0 (a 1-loop or fixed point)

1 -> 2 -> 0 (1-loop)

2 -> 0 (1-loop)

3 -> 6 -> 0 (1-loop)

4 -> 0 (1-loop)

5 -> 10 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11 (a 6-loop)

6 -> 0 (1-loop)

7 -> 14 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11 (6-loop)

8 -> 0 (1-loop)

9 -> 18 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)

10 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11 (6-loop)

11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11 (6-loop)

12 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11 (6-loop)

13 -> 17 -> 25 -> 28 -> 18 -> 11 -> 13 (6-loop)

14 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11 (6-loop)

15 -> 21 -> 20 -> 18 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)

16 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11 (6-loop)

17 -> 25 -> 28 -> 18 -> 11 -> 13 -> 17 (6-loop)

18 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)

19 -> 29 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop or fixed point)

20 -> 18 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)

21 -> 20 -> 18 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)

22 -> 18 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)

23 -> 24 -> 18 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)

24 -> 18 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)

25 -> 28 -> 18 -> 11 -> 13 -> 17 -> 25 (6-loop)

26 -> 18 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)

27 -> 32 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

28 -> 18 -> 11 -> 13 -> 17 -> 25 -> 28 (6-loop)

29 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

30 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

31 -> 35 -> 43 -> 42 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

32 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

34 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

35 -> 43 -> 42 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

37 -> 47 -> 50 -> 55 -> 65 -> 64 -> 54 -> 55 (4-loop)

38 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

40 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

41 -> 38 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

42 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

43 -> 42 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

44 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

45 -> 46 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

46 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

47 -> 50 -> 55 -> 65 -> 64 -> 54 -> 55 (4-loop)

48 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

49 -> 54 -> 55 -> 65 -> 64 -> 54 -> 55 (4-loop)

50 -> 55 -> 65 -> 64 -> 54 -> 55 (4-loop)

51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

52 -> 55 -> 65 -> 64 -> 54 -> 55 (4-loop)

53 -> 61 -> 56 -> 55 -> 65 -> 64 -> 54 -> 55 (4-loop)

54 -> 55 -> 65 -> 64 -> 54 (4-loop)

55 -> 65 -> 64 -> 54 -> 55 (4-loop)

56 -> 55 -> 65 -> 64 -> 54 -> 55 (4-loop)

57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

58 -> 55 -> 65 -> 64 -> 54 -> 55 (4-loop)

59 -> 73 -> 83 -> 78 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

60 -> 54 -> 55 -> 65 -> 64 -> 54 (4-loop)

61 -> 56 -> 55 -> 65 -> 64 -> 54 -> 55 (4-loop)

62 -> 54 -> 55 -> 65 -> 64 -> 54 (4-loop)

63 -> 60 -> 54 -> 55 -> 65 -> 64 -> 54 (4-loop)

64 -> 54 -> 55 -> 65 -> 64 (4-loop)

65 -> 64 -> 54 -> 55 -> 65 (4-loop)

66 -> 54 -> 55 -> 65 -> 64 -> 54 (4-loop)

67 -> 68 -> 54 -> 55 -> 65 -> 64 -> 54 (4-loop)

68 -> 54 -> 55 -> 65 -> 64 -> 54 (4-loop)

69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

70 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

71 -> 79 -> 95 -> 109 -> 119 -> 130 -> 134 (1-loop)

72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

73 -> 83 -> 78 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

74 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

75 -> 87 -> 86 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

76 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

78 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

79 -> 95 -> 109 -> 119 -> 130 -> 134 (1-loop)

80 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

81 -> 74 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

82 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

83 -> 78 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

84 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

85 -> 82 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

86 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

87 -> 86 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

88 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

89 -> 90 -> 99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)

90 -> 99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)

91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

92 -> 99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)

93 -> 105 -> 111 -> 114 -> 112 (1-loop)

94 -> 99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)

95 -> 108 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

96 -> 99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)

97 -> 113 -> 118 -> 112 (1-loop)

98 -> 99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)

99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)

100 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)

...

Well, is there something interesting in this exhaustive search or will those chains always end in loops and fixed points? Could all fixed points be reached? The chain 19-134 has 19 terms: could a chain be as long as one wishes? (*)

The first such fixed points are listed here: 112, 121, 134, 143, 156, 165, 178, 187, 211, 314, 336, 341, 358, 363, 385, 413, 431, 516, 538, 561, 583, 615, 633, 651, 718, 781, 817, 835, 853, 871, 1012...

The sequence of integers looping on them-selves with the rule “Add odd digits, subtract even ones” will be submitted soon to Neil Sloane’s OEIS; it starts like this (and includes all terms of A036301, of course):

0,11,13,17,18,25,28,54,55,64,65,112,121,134,137,143,148,155,156,165,166,173...

Those are the numbers highlighted in black, in the first column of the above table (and in blue on r.e.s.’s diagram at very bottom of page).

Best,

É.

(the other way rule -add even digits, subtract odd ones- is there)

_____________________________

(*) Franklin T. Adams-Watters:

(...)

Every chain of this sort will end in a loop of some sort. Consider a number N starting with n+2 “8” digits, followed by n 0's, with n >= 1. Any number less than this which transforms to something larger will still transform to number starting with n+2 8's, and then the next term(s) will be smaller until a number less than N is reached.

The chain thus cannot grow to infinity, so it must eventually loop.

Certainly the length of a chain increases without limit: start with a large enough sequence of digits with the same parity (not 0's or 9's), and you will get a large number of steps in the same direction. The interesting question is whether there are arbitrarily large loops. My guess is that there are not.

Essentially the same argument applies to adding even digits and subtracting odd; just look at 9's instead of 8's.

(...)

_____________________________

Joshua Zucker:

I played around with this a little, and found: up to 1000000, the longest cycle is 11, and cycles of that length include the following numbers (I didn't check to be sure that I have only one representative of each, though it looks like there's enough distance between them that I'm OK):

18201 81201 108201 126201 144201 162201 180201 216201 238201 261201 283201 328201 346201 364201 382201 414201 436201 441201 458201 463201 485201 548201 566201 584201 612201 621201 634201 643201 656201 665201 678201 687201 768201 786201 801201 810201 823201 832201 845201 854201 867201 876201 889201 898201 988201

Why do they all end in 201???

Here is one element of each cycle of length 10:

128201 146201 164201 182201 218201 281201 348201 366201 384201 416201 438201 461201 483201 568201 586201 614201 636201 641201 658201 663201 685201 687980 788201 812201 821201 834201 843201 856201 865201 867980 878201 887201

An example of one full cycle, length 11:

18191 -> 18195 -> 18203 -> 18197 -> 18207 -> 18205 -> 18201 -> 18193 -> 18199 -> 18211 -> 18204 -> 18191

and length 10:

128191 -> 128193 -> 128197 -> 128205 -> 128199 -> 128209 -> 128207 -> 128203 -> 128195 -> 128201 -> 128191

A cycle of length 9 that doesn't involve ending in 201 (uses 101 instead):

6095 -> 6103 -> 6101 -> 6097 -> 6107 -> 6109 -> 6113 -> 6112 -> 6106 -> 6095

and one that does:

16193 -> 16201 -> 16195 -> 16205 -> 16203 -> 16199 -> 16213 -> 16210 -> 16204 -> 16193

and here's a few really unusual ones:

600012 600005 600004 599994 600031 600029 600030 600027 600026 380037 380042 380031 380030 380028 380013 380012 380006 379995 687974 687979 687997 688015 687999 688019 688007 687992 688001 687980 867974 867979 867997 868015 867999 868019 868007 867992 868001 867980

(all the rest of length 9 or more under a million contain a number ending in 101 or 201).

Of course, with more digits there will be trivial variations of these weird ones, e.g. which replace a leading 3 with 111, or a leading 6 with 42 or 24 or 222.

_____________________________

On december 4^th, 2006, I’ve asked on rec.puzzles:

Hello,

could someone be so nice to compute for me two hundred more terms of

this sequence:

0,11,13,17,18,25,28,54,55,64,65...

Explanations can be found there:

http://www.cetteadressecomportecinquantesignes.com/134.htm

I'll mention yr name in Neil Sloane's OEIS if the sequence is accepted.

Thanks,

Éric A.

[crossposted to alt.math.recreational an hour ago]

Got this answer a few minutes later, from “Barry” and “Jongware”:

0 11 13 17 18 25 28 54 55 64 65 112 121 134 137 143 148 155 156 165 166 173

178 184 187 198 200 209 211 216 231 233 234 237 244 245 270 275 280 285 314

336 341 358 363 385 396 402 407 410 413 429 431 432 450 451 453 457 460 465

516 538 561 583 594 604 615 633 651 671 673 676 677 685 718 781 792 806 817

835 853 871 1012 1021 1034 1037 1043 1048 1055 1056 1065 1066 1073 1078 1084

1087 1102 1120 1135 1138 1145 1148 1201 1210 1217 1223 1224 1232 1235 1242

1245 1253 1254 1260 1267 1276 1289 1298 1304 1322 1340 1378 1381 1397 1403

1408 1415 1417 1418 1422 1425 1430 1447 1452 1469 1474 1496 1506 1524 1542

1560 1578 1583 1584 1595 1605 1606 1615 1616 1627 1649 1650 1672 1694 1708

1726 1744 1762 1780 1793 1804 1807 1810 1813 1829 1870 1892 1928 1946 1964

1982 2009 2011 2016 2031 2033 2034 2037 2044 2045 2070 2075 2080 2085 2101

2110 2117 2123 2124 2132 2135 2142 2145 2153 2154 2160 2167 2176 2189 2198

2207 2210 2213 2229 2231 2232 2250 2251 2253 2257 2260 2265 2312 2321 2334

2337 2343 2348 2355 2356 2365 2366 2373 2378 2384 2387 2394 2400 2415 2433

2451 2471 2473 2476 2477 2485 2514 2536 2541 2558 2563 2585 2592 2602 2617

2635 2653 2671 2716 2738 2761 2783 2790 2804 2819 2837 2855 2873 2891 2918

2981 3014 3036 3041 3058 3063 3085 3104 3122 3140 3178 3181 3212 3221 3234

3237 3243 3248 3255 3256 3265 3266 3273 3278 3284 3287 3306 3324 3342 3360

3378 3383 3384 3401 3410 3417 3423 3424 3432 3435 3442 3445 3453 3454 3460

3467 3476 3489 3498 3508 3526 3544 3562 3580 3597 3603 3608 3615 3617 3618

3621 3622 3625 3630 3647 3652 3669 3674 3696 3728 3746 3764 3782 3795 3805

3806 3812 3813 3819 3824 3827 3849 3850 3872 3894 3948 3966 3984 4007 4010

4013 4029 4031 4032 4050 4051 4053 4057 4060 4065 4097 4103 4108 4109 4115

4118 4125 4130 4147 4152 4169 4174 4196 4215 4233 4251 4271 4273 4276 4277

4285 4301 4310 4317 4323 4324 4332 4335 4342 4345 4353 4354 4360 4367 4376

4389 4398 4417 4435 4453 4471 4512 4521 4534 4537 4543 4548 4555 4556 4565

4566 4573 4578 4584 4587 4590 4600 4619 4637 4655 4673 4691 4714 4736 4741

4758 4763 4775 4785 4788 4790 4802 4839 4857 4875 4893 4916 4938 4961 4983

5016 5038 5061 5083 5106 5124 5142 5160 5178 5183 5184 5214 5236 5241 5258

5263 5285 5308 5326 5344 5362 5380 5412 5421 5434 5437 5443 5448 5455 5456

5465 5466 5473 5478 5484 5487 5528 5546 5564 5582 5601 5610 5617 5623 5624

5632 5635 5642 5645 5653 5654 5660 5667 5676 5689 5698 5748 5766 5784 5803

5825 5830 5847 5852 5869 5874 5896 5968 5986 5994 6007 6008 6011 6013 6015

6033 6051 6071 6073 6076 6077 6085 6095 6097 6101 6103 6105 6106 6107 6109

6112 6113 6127 6149 6150 6172 6194 6217 6235 6253 6271 6297 6303 6305 6307

6308 6310 6311 6325 6330 6347 6352 6369 6374 6396 6419 6437 6455 6473 6491

6501 6510 6517 6523 6524 6532 6535 6542 6545 6553 6554 6560 6567 6576 6589

6598 6599 6616 6639 6657 6675 6693 6712 6721 6734 6737 6743 6748 6755 6756

6765 6766 6773 6778 6784 6787 6797 6814 6859 6877 6895 6914 6936 6941 6958

6963 6985 7018 7081 7108 7126 7144 7162 7180 7216 7238 7261 7283 7328 7346

7364 7382 7414 7436 7441 7458 7463 7485 7548 7566 7584 7612 7621 7634 7637

7643 7648 7655 7656 7665 7666 7673 7678 7684 7687 7768 7786 7801 7810 7817

7823 7824 7832 7835 7842 7845 7853 7854 7860 7867 7876 7889 7898 7988 7998

8003 8009 8010 8013 8015 8017 8035 8053 8071 8093 8095 8097 8099 8101 8103

8105 8106 8107 8109 8111 8129 8170 8192 8219 8237 8255 8273 8291 8295 8297

8299 8301 8303 8305 8306 8307 8309 8312 8313 8327 8349 8350 8372 8394 8399

8412 8439 8457 8475 8493 8497 8499 8501 8503 8505 8507 8508 8510 8511 8525

8530 8547 8552 8569 8574 8596 8597 8610 8659 8677 8695 8701 8710 8717 8723

8724 8732 8735 8742 8745 8753 8754 8760 8767 8776 8789 8798 8879 8897 8912

8921 8934 8937 8943 8948 8955 8956 8965 8966 8973 8978 8984 8987 9128 9146

9164 9182 9218 9281 9348 9366 9384 9416 9438 9461 9483 9568 9586 9614 9636

9641 9658 9663 9685 9788 9812 9821 9834 9837 9843 9848 9855 9856 9865 9866

9873 9878 9884 9887 10012 10021 10034 10037 10043 10048 10055 10056 10065

10066 10073 10078 10084 10087 10102 10120 10135 10138 10145 10148 10201

10210 10217 10223 10224 10232 10235 10242 10245 10253 10254 10260 10267

10276 10289 10298 10304 10322 10340 10378 10381 10397 10403 10408 10415

10417 10418 10422 10425 10430 10447 10452 10469 10474 10496 10506 10524

10542 10560 10578 10583 10584 10595 10605 10606 10615 10616 10627 10649

10650 10672 10694 10708 10726 10744 10762 10780 10793 10804 10807 10810

10813 10829 10870 10892 10928 10946 10964 10982 11002 11020 11035 11038

11045 11048 11114 11136 11141 11158 11163 11185 11200 11211 11213 11217

11218 11225 11228 11254 11255 11264 11265 11316 11338 11361 11383 11398

11404 11409 11411 11416 11431 11433 11434 11437 11444 11445 11470 11475

11480 11485 11518 11581 11596 11606 11607 11610 11613 11629 11631 11632

11650 11651 11653 11657 11660 11665 11794 11808 11815 11833 11851 11871

11873 11876 11877 11885 12001 12010 12017 12023 12024 12032 12035 12042

12045 12053 12054 12060 12067 12076 12089 12098 12100 12111 12113 12117

12118 12125 12128 12154 12155 12164 12165 12197 12203 12208 12213 12214

12225 12230 12247 12252 12269 12274 12296 12302 12320 12335 12338 12345

12348 12395 12405 12408 12411 12427 12449 12450 12472 12494 12504 12522

12540 12578 12581 12593 12606 12607 12609 12611 12629 12670 12692 12706

12724 12742 12760 12778 12783 12784 12791 12793 12801 12804 12805 12807

12809 12811 12890 12908 12926 12944 12962 12980 13004 13022 13040 13078

13081 13116 13138 13161 13183 13202 13220 13235 13238 13245 13248 13318

13381 13400 13411 13413 13417 13418 13425 13428 13454 13455 13464 13465

13598 13608 13609 13611 13616 13631 13633 13634 13637 13644 13645 13670

13675 13680 13685 13807 13810 13813 13829 13831 13832 13850 13851 13853

13857 13860 13865 13997 14003 14008 14015 14018 14025 14026 14030 14047

14052 14069 14074 14096 14109 14111 14116 14131 14133 14134 14137 14144

14145 14170 14175 14180 14185 14195 14205 14206 14207 14209 14212 14213

14227 14249 14250 14272 14294 14300 14311 14313 14317 14318 14325 14328

14354 14355 14364 14365 14393 14399 14403 14404 14405 14407 14417 14418

14429 14470 14492 14502 14520 14535 14538 14545 14548 14591 14593 14597

14601 14602 14603 14605 14607 14609 14612 14615 14690 14704 14722 14740

14778 14781 14789 14794 14795 14803 14806 14813 14906 14924 14942 14960

14978 14983 14984 15006 15024 15042 15060 15078 15083 15084 15118 15181

15204 15222 15240 15278 15281 15402 15420 15435 15438 15445 15448 15600

15611 15613 15617 15618 15625 15628 15654 15655 15664 15665 15809 15811

15816 15831 15833 15834 15837 15844 15845 15870 15875 15880 15885 15995

16005 16006 16012 16013 16024 16027 16049 16050 16072 16094 16107 16110