Biquanimous Numbers

The number associated with my diurnal age today, 27190, has the interesting property that it counts the number of five digit biquanimous numbers. This covers the range from 10000 to 99999 which means that about 27% of five digit numbers are biquanimous, a term used to describe numbers whose digits can be split into two groups with equal sums. By an odd coincidence, the number following 27190 is an example of such a number because:$$27191 \rightarrow \underbrace{2+7+1}_{\text{ sums to } 10} \text{ and } \underbrace{9+1}_{\text{ sums to } 10}$$Now just to repeat: 27190 is not biquanimous but it does count the number of five digit biquanimous numbers. These totals of \(n\)-digit biquanimous numbers constitute OEIS A065086:

A065086

Number of \(n\)-digit biquanimous numbers in base 10 not allowing leading zeros.

The initial members of the sequence are:

1, 9, 126, 1920, 27190, 347168, 3990467, 42744527, 440764556, 4464045276, 44863859589, 449488519847, 4498059105204, 44992451829730, 449969622539954, 4499873022468708, 44999449703306768, 449997540235466340, 4499988731927483569, 44999947410278947807

It can be seen that the limit as \(n \rightarrow \infty \) seems to be 45% but in fact, if leading zeros are allowed, the limit is 0.5 according to this source. The biquanimous numbers, also known as biquams, are listed in OEIS A064544:

A064544

Biquanimous numbers (or biquams): group the digits into two pieces (not necessarily equal or in order) with the same sum.

The initial members are:

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 112, 121, 123, 132, 134, 143, 145, 154, 156, 165, 167, 176, 178, 187, 189, 198, 202, 211, 213, 220, 224, 231, 235, 242, 246, 253, 257, 264, 268, 275, 279, 286, 297, 303, 312, 314, 321, 325, 330, 336, 341, 347, 352, 358

This approach is similar to totaling the odd and even numbers and seeing if they balance. Numbers that do constitute OEIS  A036301:

A036301

Numbers whose sum of even digits and sum of odd digits are equal.

The initial members are:

0, 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, 1021, 1034, 1043, 1056, 1065, 1078, 1087, 1102, 1120, 1201, 1210, 1223, 1232, 1245, 1254, 1267, 1276, 1289, 1298 

It can be noted that all such numbers are by definition biquanimous because their digits can be split into two groups with equal sums. For example:$$1298 \rightarrow \underbrace{1+9}_{\text{sums to }10 } \text{ and } \underbrace{2+8}_{ \text{sums to } 10}$$

Balanced Numbers

My diurnal age today is 27018 and the sum of the digits to the left of the zero are equal to the sum of the digits to the right of the zero (both total 9):$$ 27018=\overbrace{27}^{2+7=9} \cdot 0 \cdot \overbrace{18}^{1+8=9}$$Such a number is often referred to as a balanced number. If the number has an even number of digits, then there is no middle digit. An example is 2341:$$ 2341 = \overbrace{23}^{2+3=5} \cdot \overbrace{41}^{4+1=5}$$It's a simple enough concept and it's not difficult to write some SageMath code to find all such numbers between 10 and 40000. Here's the permalink. I developed this code myself but Geeks for Geeks has the code in C++, Java, Python3, C# and Javascript.

All palindromic numbers are of course balanced numbers. Here are the 2764 balanced numbers in the range, constituting 6.91% of the total:

11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 707, 717, 727, 737, 747, 757, 767, 777, 787, 797, 808, 818, 828, 838, 848, 858, 868, 878, 888, 898, 909, 919, 929, 939, 949, 959, 969, 979, 989, 999, 1001, 1010, 1102, 1111, 1120, 1203, 1212, 1221, 1230, 1304, 1313, 1322, 1331, 1340, 1405, 1414, 1423, 1432, 1441, 1450, 1506, 1515, 1524, 1533, 1542, 1551, 1560, 1607, 1616, 1625, 1634, 1643, 1652, 1661, 1670, 1708, 1717, 1726, 1735, 1744, 1753, 1762, 1771, 1780, 1809, 1818, 1827, 1836, 1845, 1854, 1863, 1872, 1881, 1890, 1919, 1928, 1937, 1946, 1955, 1964, 1973, 1982, 1991, 2002, 2011, 2020, 2103, 2112, 2121, 2130, 2204, 2213, 2222, 2231, 2240, 2305, 2314, 2323, 2332, 2341, 2350, 2406, 2415, 2424, 2433, 2442, 2451, 2460, 2507, 2516, 2525, 2534, 2543, 2552, 2561, 2570, 2608, 2617, 2626, 2635, 2644, 2653, 2662, 2671, 2680, 2709, 2718, 2727, 2736, 2745, 2754, 2763, 2772, 2781, 2790, 2819, 2828, 2837, 2846, 2855, 2864, 2873, 2882, 2891, 2929, 2938, 2947, 2956, 2965, 2974, 2983, 2992, 3003, 3012, 3021, 3030, 3104, 3113, 3122, 3131, 3140, 3205, 3214, 3223, 3232, 3241, 3250, 3306, 3315, 3324, 3333, 3342, 3351, 3360, 3407, 3416, 3425, 3434, 3443, 3452, 3461, 3470, 3508, 3517, 3526, 3535, 3544, 3553, 3562, 3571, 3580, 3609, 3618, 3627, 3636, 3645, 3654, 3663, 3672, 3681, 3690, 3719, 3728, 3737, 3746, 3755, 3764, 3773, 3782, 3791, 3829, 3838, 3847, 3856, 3865, 3874, 3883, 3892, 3939, 3948, 3957, 3966, 3975, 3984, 3993, 4004, 4013, 4022, 4031, 4040, 4105, 4114, 4123, 4132, 4141, 4150, 4206, 4215, 4224, 4233, 4242, 4251, 4260, 4307, 4316, 4325, 4334, 4343, 4352, 4361, 4370, 4408, 4417, 4426, 4435, 4444, 4453, 4462, 4471, 4480, 4509, 4518, 4527, 4536, 4545, 4554, 4563, 4572, 4581, 4590, 4619, 4628, 4637, 4646, 4655, 4664, 4673, 4682, 4691, 4729, 4738, 4747, 4756, 4765, 4774, 4783, 4792, 4839, 4848, 4857, 4866, 4875, 4884, 4893, 4949, 4958, 4967, 4976, 4985, 4994, 5005, 5014, 5023, 5032, 5041, 5050, 5106, 5115, 5124, 5133, 5142, 5151, 5160, 5207, 5216, 5225, 5234, 5243, 5252, 5261, 5270, 5308, 5317, 5326, 5335, 5344, 5353, 5362, 5371, 5380, 5409, 5418, 5427, 5436, 5445, 5454, 5463, 5472, 5481, 5490, 5519, 5528, 5537, 5546, 5555, 5564, 5573, 5582, 5591, 5629, 5638, 5647, 5656, 5665, 5674, 5683, 5692, 5739, 5748, 5757, 5766, 5775, 5784, 5793, 5849, 5858, 5867, 5876, 5885, 5894, 5959, 5968, 5977, 5986, 5995, 6006, 6015, 6024, 6033, 6042, 6051, 6060, 6107, 6116, 6125, 6134, 6143, 6152, 6161, 6170, 6208, 6217, 6226, 6235, 6244, 6253, 6262, 6271, 6280, 6309, 6318, 6327, 6336, 6345, 6354, 6363, 6372, 6381, 6390, 6419, 6428, 6437, 6446, 6455, 6464, 6473, 6482, 6491, 6529, 6538, 6547, 6556, 6565, 6574, 6583, 6592, 6639, 6648, 6657, 6666, 6675, 6684, 6693, 6749, 6758, 6767, 6776, 6785, 6794, 6859, 6868, 6877, 6886, 6895, 6969, 6978, 6987, 6996, 7007, 7016, 7025, 7034, 7043, 7052, 7061, 7070, 7108, 7117, 7126, 7135, 7144, 7153, 7162, 7171, 7180, 7209, 7218, 7227, 7236, 7245, 7254, 7263, 7272, 7281, 7290, 7319, 7328, 7337, 7346, 7355, 7364, 7373, 7382, 7391, 7429, 7438, 7447, 7456, 7465, 7474, 7483, 7492, 7539, 7548, 7557, 7566, 7575, 7584, 7593, 7649, 7658, 7667, 7676, 7685, 7694, 7759, 7768, 7777, 7786, 7795, 7869, 7878, 7887, 7896, 7979, 7988, 7997, 8008, 8017, 8026, 8035, 8044, 8053, 8062, 8071, 8080, 8109, 8118, 8127, 8136, 8145, 8154, 8163, 8172, 8181, 8190, 8219, 8228, 8237, 8246, 8255, 8264, 8273, 8282, 8291, 8329, 8338, 8347, 8356, 8365, 8374, 8383, 8392, 8439, 8448, 8457, 8466, 8475, 8484, 8493, 8549, 8558, 8567, 8576, 8585, 8594, 8659, 8668, 8677, 8686, 8695, 8769, 8778, 8787, 8796, 8879, 8888, 8897, 8989, 8998, 9009, 9018, 9027, 9036, 9045, 9054, 9063, 9072, 9081, 9090, 9119, 9128, 9137, 9146, 9155, 9164, 9173, 9182, 9191, 9229, 9238, 9247, 9256, 9265, 9274, 9283, 9292, 9339, 9348, 9357, 9366, 9375, 9384, 9393, 9449, 9458, 9467, 9476, 9485, 9494, 9559, 9568, 9577, 9586, 9595, 9669, 9678, 9687, 9696, 9779, 9788, 9797, 9889, 9898, 9999, 10001, 10010, 10101, 10110, 10201, 10210, 10301, 10310, 10401, 10410, 10501, 10510, 10601, 10610, 10701, 10710, 10801, 10810, 10901, 10910, 11002, 11011, 11020, 11102, 11111, 11120, 11202, 11211, 11220, 11302, 11311, 11320, 11402, 11411, 11420, 11502, 11511, 11520, 11602, 11611, 11620, 11702, 11711, 11720, 11802, 11811, 11820, 11902, 11911, 11920, 12003, 12012, 12021, 12030, 12103, 12112, 12121, 12130, 12203, 12212, 12221, 12230, 12303, 12312, 12321, 12330, 12403, 12412, 12421, 12430, 12503, 12512, 12521, 12530, 12603, 12612, 12621, 12630, 12703, 12712, 12721, 12730, 12803, 12812, 12821, 12830, 12903, 12912, 12921, 12930, 13004, 13013, 13022, 13031, 13040, 13104, 13113, 13122, 13131, 13140, 13204, 13213, 13222, 13231, 13240, 13304, 13313, 13322, 13331, 13340, 13404, 13413, 13422, 13431, 13440, 13504, 13513, 13522, 13531, 13540, 13604, 13613, 13622, 13631, 13640, 13704, 13713, 13722, 13731, 13740, 13804, 13813, 13822, 13831, 13840, 13904, 13913, 13922, 13931, 13940, 14005, 14014, 14023, 14032, 14041, 14050, 14105, 14114, 14123, 14132, 14141, 14150, 14205, 14214, 14223, 14232, 14241, 14250, 14305, 14314, 14323, 14332, 14341, 14350, 14405, 14414, 14423, 14432, 14441, 14450, 14505, 14514, 14523, 14532, 14541, 14550, 14605, 14614, 14623, 14632, 14641, 14650, 14705, 14714, 14723, 14732, 14741, 14750, 14805, 14814, 14823, 14832, 14841, 14850, 14905, 14914, 14923, 14932, 14941, 14950, 15006, 15015, 15024, 15033, 15042, 15051, 15060, 15106, 15115, 15124, 15133, 15142, 15151, 15160, 15206, 15215, 15224, 15233, 15242, 15251, 15260, 15306, 15315, 15324, 15333, 15342, 15351, 15360, 15406, 15415, 15424, 15433, 15442, 15451, 15460, 15506, 15515, 15524, 15533, 15542, 15551, 15560, 15606, 15615, 15624, 15633, 15642, 15651, 15660, 15706, 15715, 15724, 15733, 15742, 15751, 15760, 15806, 15815, 15824, 15833, 15842, 15851, 15860, 15906, 15915, 15924, 15933, 15942, 15951, 15960, 16007, 16016, 16025, 16034, 16043, 16052, 16061, 16070, 16107, 16116, 16125, 16134, 16143, 16152, 16161, 16170, 16207, 16216, 16225, 16234, 16243, 16252, 16261, 16270, 16307, 16316, 16325, 16334, 16343, 16352, 16361, 16370, 16407, 16416, 16425, 16434, 16443, 16452, 16461, 16470, 16507, 16516, 16525, 16534, 16543, 16552, 16561, 16570, 16607, 16616, 16625, 16634, 16643, 16652, 16661, 16670, 16707, 16716, 16725, 16734, 16743, 16752, 16761, 16770, 16807, 16816, 16825, 16834, 16843, 16852, 16861, 16870, 16907, 16916, 16925, 16934, 16943, 16952, 16961, 16970, 17008, 17017, 17026, 17035, 17044, 17053, 17062, 17071, 17080, 17108, 17117, 17126, 17135, 17144, 17153, 17162, 17171, 17180, 17208, 17217, 17226, 17235, 17244, 17253, 17262, 17271, 17280, 17308, 17317, 17326, 17335, 17344, 17353, 17362, 17371, 17380, 17408, 17417, 17426, 17435, 17444, 17453, 17462, 17471, 17480, 17508, 17517, 17526, 17535, 17544, 17553, 17562, 17571, 17580, 17608, 17617, 17626, 17635, 17644, 17653, 17662, 17671, 17680, 17708, 17717, 17726, 17735, 17744, 17753, 17762, 17771, 17780, 17808, 17817, 17826, 17835, 17844, 17853, 17862, 17871, 17880, 17908, 17917, 17926, 17935, 17944, 17953, 17962, 17971, 17980, 18009, 18018, 18027, 18036, 18045, 18054, 18063, 18072, 18081, 18090, 18109, 18118, 18127, 18136, 18145, 18154, 18163, 18172, 18181, 18190, 18209, 18218, 18227, 18236, 18245, 18254, 18263, 18272, 18281, 18290, 18309, 18318, 18327, 18336, 18345, 18354, 18363, 18372, 18381, 18390, 18409, 18418, 18427, 18436, 18445, 18454, 18463, 18472, 18481, 18490, 18509, 18518, 18527, 18536, 18545, 18554, 18563, 18572, 18581, 18590, 18609, 18618, 18627, 18636, 18645, 18654, 18663, 18672, 18681, 18690, 18709, 18718, 18727, 18736, 18745, 18754, 18763, 18772, 18781, 18790, 18809, 18818, 18827, 18836, 18845, 18854, 18863, 18872, 18881, 18890, 18909, 18918, 18927, 18936, 18945, 18954, 18963, 18972, 18981, 18990, 19019, 19028, 19037, 19046, 19055, 19064, 19073, 19082, 19091, 19119, 19128, 19137, 19146, 19155, 19164, 19173, 19182, 19191, 19219, 19228, 19237, 19246, 19255, 19264, 19273, 19282, 19291, 19319, 19328, 19337, 19346, 19355, 19364, 19373, 19382, 19391, 19419, 19428, 19437, 19446, 19455, 19464, 19473, 19482, 19491, 19519, 19528, 19537, 19546, 19555, 19564, 19573, 19582, 19591, 19619, 19628, 19637, 19646, 19655, 19664, 19673, 19682, 19691, 19719, 19728, 19737, 19746, 19755, 19764, 19773, 19782, 19791, 19819, 19828, 19837, 19846, 19855, 19864, 19873, 19882, 19891, 19919, 19928, 19937, 19946, 19955, 19964, 19973, 19982, 19991, 20002, 20011, 20020, 20102, 20111, 20120, 20202, 20211, 20220, 20302, 20311, 20320, 20402, 20411, 20420, 20502, 20511, 20520, 20602, 20611, 20620, 20702, 20711, 20720, 20802, 20811, 20820, 20902, 20911, 20920, 21003, 21012, 21021, 21030, 21103, 21112, 21121, 21130, 21203, 21212, 21221, 21230, 21303, 21312, 21321, 21330, 21403, 21412, 21421, 21430, 21503, 21512, 21521, 21530, 21603, 21612, 21621, 21630, 21703, 21712, 21721, 21730, 21803, 21812, 21821, 21830, 21903, 21912, 21921, 21930, 22004, 22013, 22022, 22031, 22040, 22104, 22113, 22122, 22131, 22140, 22204, 22213, 22222, 22231, 22240, 22304, 22313, 22322, 22331, 22340, 22404, 22413, 22422, 22431, 22440, 22504, 22513, 22522, 22531, 22540, 22604, 22613, 22622, 22631, 22640, 22704, 22713, 22722, 22731, 22740, 22804, 22813, 22822, 22831, 22840, 22904, 22913, 22922, 22931, 22940, 23005, 23014, 23023, 23032, 23041, 23050, 23105, 23114, 23123, 23132, 23141, 23150, 23205, 23214, 23223, 23232, 23241, 23250, 23305, 23314, 23323, 23332, 23341, 23350, 23405, 23414, 23423, 23432, 23441, 23450, 23505, 23514, 23523, 23532, 23541, 23550, 23605, 23614, 23623, 23632, 23641, 23650, 23705, 23714, 23723, 23732, 23741, 23750, 23805, 23814, 23823, 23832, 23841, 23850, 23905, 23914, 23923, 23932, 23941, 23950, 24006, 24015, 24024, 24033, 24042, 24051, 24060, 24106, 24115, 24124, 24133, 24142, 24151, 24160, 24206, 24215, 24224, 24233, 24242, 24251, 24260, 24306, 24315, 24324, 24333, 24342, 24351, 24360, 24406, 24415, 24424, 24433, 24442, 24451, 24460, 24506, 24515, 24524, 24533, 24542, 24551, 24560, 24606, 24615, 24624, 24633, 24642, 24651, 24660, 24706, 24715, 24724, 24733, 24742, 24751, 24760, 24806, 24815, 24824, 24833, 24842, 24851, 24860, 24906, 24915, 24924, 24933, 24942, 24951, 24960, 25007, 25016, 25025, 25034, 25043, 25052, 25061, 25070, 25107, 25116, 25125, 25134, 25143, 25152, 25161, 25170, 25207, 25216, 25225, 25234, 25243, 25252, 25261, 25270, 25307, 25316, 25325, 25334, 25343, 25352, 25361, 25370, 25407, 25416, 25425, 25434, 25443, 25452, 25461, 25470, 25507, 25516, 25525, 25534, 25543, 25552, 25561, 25570, 25607, 25616, 25625, 25634, 25643, 25652, 25661, 25670, 25707, 25716, 25725, 25734, 25743, 25752, 25761, 25770, 25807, 25816, 25825, 25834, 25843, 25852, 25861, 25870, 25907, 25916, 25925, 25934, 25943, 25952, 25961, 25970, 26008, 26017, 26026, 26035, 26044, 26053, 26062, 26071, 26080, 26108, 26117, 26126, 26135, 26144, 26153, 26162, 26171, 26180, 26208, 26217, 26226, 26235, 26244, 26253, 26262, 26271, 26280, 26308, 26317, 26326, 26335, 26344, 26353, 26362, 26371, 26380, 26408, 26417, 26426, 26435, 26444, 26453, 26462, 26471, 26480, 26508, 26517, 26526, 26535, 26544, 26553, 26562, 26571, 26580, 26608, 26617, 26626, 26635, 26644, 26653, 26662, 26671, 26680, 26708, 26717, 26726, 26735, 26744, 26753, 26762, 26771, 26780, 26808, 26817, 26826, 26835, 26844, 26853, 26862, 26871, 26880, 26908, 26917, 26926, 26935, 26944, 26953, 26962, 26971, 26980, 27009, 27018, 27027, 27036, 27045, 27054, 27063, 27072, 27081, 27090, 27109, 27118, 27127, 27136, 27145, 27154, 27163, 27172, 27181, 27190, 27209, 27218, 27227, 27236, 27245, 27254, 27263, 27272, 27281, 27290, 27309, 27318, 27327, 27336, 27345, 27354, 27363, 27372, 27381, 27390, 27409, 27418, 27427, 27436, 27445, 27454, 27463, 27472, 27481, 27490, 27509, 27518, 27527, 27536, 27545, 27554, 27563, 27572, 27581, 27590, 27609, 27618, 27627, 27636, 27645, 27654, 27663, 27672, 27681, 27690, 27709, 27718, 27727, 27736, 27745, 27754, 27763, 27772, 27781, 27790, 27809, 27818, 27827, 27836, 27845, 27854, 27863, 27872, 27881, 27890, 27909, 27918, 27927, 27936, 27945, 27954, 27963, 27972, 27981, 27990, 28019, 28028, 28037, 28046, 28055, 28064, 28073, 28082, 28091, 28119, 28128, 28137, 28146, 28155, 28164, 28173, 28182, 28191, 28219, 28228, 28237, 28246, 28255, 28264, 28273, 28282, 28291, 28319, 28328, 28337, 28346, 28355, 28364, 28373, 28382, 28391, 28419, 28428, 28437, 28446, 28455, 28464, 28473, 28482, 28491, 28519, 28528, 28537, 28546, 28555, 28564, 28573, 28582, 28591, 28619, 28628, 28637, 28646, 28655, 28664, 28673, 28682, 28691, 28719, 28728, 28737, 28746, 28755, 28764, 28773, 28782, 28791, 28819, 28828, 28837, 28846, 28855, 28864, 28873, 28882, 28891, 28919, 28928, 28937, 28946, 28955, 28964, 28973, 28982, 28991, 29029, 29038, 29047, 29056, 29065, 29074, 29083, 29092, 29129, 29138, 29147, 29156, 29165, 29174, 29183, 29192, 29229, 29238, 29247, 29256, 29265, 29274, 29283, 29292, 29329, 29338, 29347, 29356, 29365, 29374, 29383, 29392, 29429, 29438, 29447, 29456, 29465, 29474, 29483, 29492, 29529, 29538, 29547, 29556, 29565, 29574, 29583, 29592, 29629, 29638, 29647, 29656, 29665, 29674, 29683, 29692, 29729, 29738, 29747, 29756, 29765, 29774, 29783, 29792, 29829, 29838, 29847, 29856, 29865, 29874, 29883, 29892, 29929, 29938, 29947, 29956, 29965, 29974, 29983, 29992, 30003, 30012, 30021, 30030, 30103, 30112, 30121, 30130, 30203, 30212, 30221, 30230, 30303, 30312, 30321, 30330, 30403, 30412, 30421, 30430, 30503, 30512, 30521, 30530, 30603, 30612, 30621, 30630, 30703, 30712, 30721, 30730, 30803, 30812, 30821, 30830, 30903, 30912, 30921, 30930, 31004, 31013, 31022, 31031, 31040, 31104, 31113, 31122, 31131, 31140, 31204, 31213, 31222, 31231, 31240, 31304, 31313, 31322, 31331, 31340, 31404, 31413, 31422, 31431, 31440, 31504, 31513, 31522, 31531, 31540, 31604, 31613, 31622, 31631, 31640, 31704, 31713, 31722, 31731, 31740, 31804, 31813, 31822, 31831, 31840, 31904, 31913, 31922, 31931, 31940, 32005, 32014, 32023, 32032, 32041, 32050, 32105, 32114, 32123, 32132, 32141, 32150, 32205, 32214, 32223, 32232, 32241, 32250, 32305, 32314, 32323, 32332, 32341, 32350, 32405, 32414, 32423, 32432, 32441, 32450, 32505, 32514, 32523, 32532, 32541, 32550, 32605, 32614, 32623, 32632, 32641, 32650, 32705, 32714, 32723, 32732, 32741, 32750, 32805, 32814, 32823, 32832, 32841, 32850, 32905, 32914, 32923, 32932, 32941, 32950, 33006, 33015, 33024, 33033, 33042, 33051, 33060, 33106, 33115, 33124, 33133, 33142, 33151, 33160, 33206, 33215, 33224, 33233, 33242, 33251, 33260, 33306, 33315, 33324, 33333, 33342, 33351, 33360, 33406, 33415, 33424, 33433, 33442, 33451, 33460, 33506, 33515, 33524, 33533, 33542, 33551, 33560, 33606, 33615, 33624, 33633, 33642, 33651, 33660, 33706, 33715, 33724, 33733, 33742, 33751, 33760, 33806, 33815, 33824, 33833, 33842, 33851, 33860, 33906, 33915, 33924, 33933, 33942, 33951, 33960, 34007, 34016, 34025, 34034, 34043, 34052, 34061, 34070, 34107, 34116, 34125, 34134, 34143, 34152, 34161, 34170, 34207, 34216, 34225, 34234, 34243, 34252, 34261, 34270, 34307, 34316, 34325, 34334, 34343, 34352, 34361, 34370, 34407, 34416, 34425, 34434, 34443, 34452, 34461, 34470, 34507, 34516, 34525, 34534, 34543, 34552, 34561, 34570, 34607, 34616, 34625, 34634, 34643, 34652, 34661, 34670, 34707, 34716, 34725, 34734, 34743, 34752, 34761, 34770, 34807, 34816, 34825, 34834, 34843, 34852, 34861, 34870, 34907, 34916, 34925, 34934, 34943, 34952, 34961, 34970, 35008, 35017, 35026, 35035, 35044, 35053, 35062, 35071, 35080, 35108, 35117, 35126, 35135, 35144, 35153, 35162, 35171, 35180, 35208, 35217, 35226, 35235, 35244, 35253, 35262, 35271, 35280, 35308, 35317, 35326, 35335, 35344, 35353, 35362, 35371, 35380, 35408, 35417, 35426, 35435, 35444, 35453, 35462, 35471, 35480, 35508, 35517, 35526, 35535, 35544, 35553, 35562, 35571, 35580, 35608, 35617, 35626, 35635, 35644, 35653, 35662, 35671, 35680, 35708, 35717, 35726, 35735, 35744, 35753, 35762, 35771, 35780, 35808, 35817, 35826, 35835, 35844, 35853, 35862, 35871, 35880, 35908, 35917, 35926, 35935, 35944, 35953, 35962, 35971, 35980, 36009, 36018, 36027, 36036, 36045, 36054, 36063, 36072, 36081, 36090, 36109, 36118, 36127, 36136, 36145, 36154, 36163, 36172, 36181, 36190, 36209, 36218, 36227, 36236, 36245, 36254, 36263, 36272, 36281, 36290, 36309, 36318, 36327, 36336, 36345, 36354, 36363, 36372, 36381, 36390, 36409, 36418, 36427, 36436, 36445, 36454, 36463, 36472, 36481, 36490, 36509, 36518, 36527, 36536, 36545, 36554, 36563, 36572, 36581, 36590, 36609, 36618, 36627, 36636, 36645, 36654, 36663, 36672, 36681, 36690, 36709, 36718, 36727, 36736, 36745, 36754, 36763, 36772, 36781, 36790, 36809, 36818, 36827, 36836, 36845, 36854, 36863, 36872, 36881, 36890, 36909, 36918, 36927, 36936, 36945, 36954, 36963, 36972, 36981, 36990, 37019, 37028, 37037, 37046, 37055, 37064, 37073, 37082, 37091, 37119, 37128, 37137, 37146, 37155, 37164, 37173, 37182, 37191, 37219, 37228, 37237, 37246, 37255, 37264, 37273, 37282, 37291, 37319, 37328, 37337, 37346, 37355, 37364, 37373, 37382, 37391, 37419, 37428, 37437, 37446, 37455, 37464, 37473, 37482, 37491, 37519, 37528, 37537, 37546, 37555, 37564, 37573, 37582, 37591, 37619, 37628, 37637, 37646, 37655, 37664, 37673, 37682, 37691, 37719, 37728, 37737, 37746, 37755, 37764, 37773, 37782, 37791, 37819, 37828, 37837, 37846, 37855, 37864, 37873, 37882, 37891, 37919, 37928, 37937, 37946, 37955, 37964, 37973, 37982, 37991, 38029, 38038, 38047, 38056, 38065, 38074, 38083, 38092, 38129, 38138, 38147, 38156, 38165, 38174, 38183, 38192, 38229, 38238, 38247, 38256, 38265, 38274, 38283, 38292, 38329, 38338, 38347, 38356, 38365, 38374, 38383, 38392, 38429, 38438, 38447, 38456, 38465, 38474, 38483, 38492, 38529, 38538, 38547, 38556, 38565, 38574, 38583, 38592, 38629, 38638, 38647, 38656, 38665, 38674, 38683, 38692, 38729, 38738, 38747, 38756, 38765, 38774, 38783, 38792, 38829, 38838, 38847, 38856, 38865, 38874, 38883, 38892, 38929, 38938, 38947, 38956, 38965, 38974, 38983, 38992, 39039, 39048, 39057, 39066, 39075, 39084, 39093, 39139, 39148, 39157, 39166, 39175, 39184, 39193, 39239, 39248, 39257, 39266, 39275, 39284, 39293, 39339, 39348, 39357, 39366, 39375, 39384, 39393, 39439, 39448, 39457, 39466, 39475, 39484, 39493, 39539, 39548, 39557, 39566, 39575, 39584, 39593, 39639, 39648, 39657, 39666, 39675, 39684, 39693, 39739, 39748, 39757, 39766, 39775, 39784, 39793, 39839, 39848, 39857, 39866, 39875, 39884, 39893, 39939, 39948, 39957, 39966, 39975, 39984, 39993