Balanced numbers

L’Aventurier, Roland Topor

The palindromes 66 and 101, for instance, are balanced numbers: if you put them on a graduated ruler which is placed on a fulcrum, the ruler will remain horizontal.

But there are balanced numbers which are not palindromes – like 102 or 2024:

Why are 102 and 2024 balanced numbers? Because the distance which separates a digit from the fulcrum comes here into play: the more this distance increases, the heavier the digit in question is.

We see above that the digit 1 of 102 weighs 2 in reality (weight*distance = 1*2 = 2). This quantity balances the influence of the 2 of 102 (for which weight*distance = 2*1 = 2 too).

We have the same kind of balance with the number 2024 if we put the fulcrum exactly under the rightmost 2.

I am 99% sure that those balanced numbers are in the OEIS — but I could not find them: where are they?

And, a last little puzzle: can you find the first four consecutive balanced numbers > 9 [this is a(n), a(n+1), a(n+2), a(n+3)]?

____________________

Next day update

Giorgos Kalogeropoulos was quick to send me this:

>  Eric, I think you are looking for A282151.

ÉA

Well spotted, Giorgos, thanks! I guess that the above seq you found, together with A282115, cover all fulcrum's possible places mentioned on this page.

As for the puzzle, I have a small doubt about my answer – which was intended to be {99, 100, 101, 102}.

I doubt because 100 is not in the last OEIS seq by Paolo Lava– nor are 200, 300, 400,... 

Similarly, 10, 20, 30, 40,...  or 1000, 2000, 3000, 4000,...  are not in the first OEIS seq by Paolo. What do you think?

GK

Actually, the way I understood the puzzle was that we should have at least one digit on each side. This is why I get the same results with the OEIS seqs. But, as always, it is your puzzle - your rules ;-)

ÉA

Good answer, Giorgos! 

I propose to submit to readers the sequence B of the 2222 first nonnegative balanced integers, using the b-file computed by Hans Havermann here — this means that we authorize ourselves to put the fulcrum under the leftmost digit of some terms: 

B= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 102, 110, 111, 113, 121, 124, 131, 135, 141, 146, 151, 157, 161, 168, 171, 179, 181, 191, 200, 201, 202, 204, 212, 215, 220, 222, 226, 232, 237, 242, 248, 252, 259, 262, 272, 282, 292, 300, 303, 306, 311, 313, 317, 323, 328, 330, 333, 339, 343, 353, 363, 373, 383, 393, 400, 402, 404, 408, 414, 419, 421, 424, 434, 440, 444, 454, 464, 474, 484, 494, 500, 505, 512, 515, 525, 531, 535, 545, 550, 555, 565, 575, 585, 595, 600, 603, 606, 616, 622, 626, 636, 641, 646, 656, 660, 666, 676, 686, 696, 700, 707, 713, 717, 727, 732, 737, 747, 751, 757, 767, 770, 777, 787, 797, 800, 804, 808, 818, 823, 828, 838, 842, 848, 858, 861, 868, 878, 880, 888, 898, 900, 909, 914, 919, 929, 933, 939, 949, 952, 959, 969, 971, 979, 989, 990, 999, 1000, 1001, 1002, 1003, 1010, 1012, 1014, 1020, 1022, 1025, 1032, 1036, 1042, 1047, 1052, 1058, 1062, 1069, 1072, 1082, 1092, 1100, 1103, 1105, 1110, 1111, 1113, 1116, 1123, 1127, 1130, 1133, 1138, 1143, 1149, 1153, 1163, 1173, 1183, 1193, 1202, 1204, 1207, 1210, 1214, 1218, 1221, 1224, 1229, 1234, 1240, 1244, 1254, 1264, 1274, 1284, 1294, 1305, 1309, 1310, 1312, 1315, 1325, 1331, 1335, 1345, 1350, 1355, 1365, 1375, 1385, 1395, 1403, 1406, 1410, 1416, 1422, 1426, 1436, 1441, 1446, 1456, 1460, 1466, 1476, 1486, 1496, 1507, 1510, 1513, 1517, 1527, 1532, 1537, 1547, 1551, 1557, 1567, 1570, 1577, 1587, 1597, 1604, 1608, 1610, 1618, 1623, 1628, 1638, 1642, 1648, 1658, 1661, 1668, 1678, 1680, 1688, 1698, 1709, 1710, 1714, 1719, 1729, 1733, 1739, 1749, 1752, 1759, 1769, 1771, 1779, 1789, 1790, 1799, 1805, 1810, 1824, 1843, 1862, 1881, 1910, 1915, 1934, 1953, 1972, 1991, 2000, 2001, 2002, 2004, 2006, 2010, 2014, 2017, 2020, 2021, 2024, 2028, 2034, 2039, 2040, 2044, 2054, 2064, 2074, 2084, 2094, 2101, 2105, 2108, 2112, 2115, 2119, 2120, 2125, 2131, 2135, 2145, 2150, 2155, 2165, 2175, 2185, 2195, 2200, 2201, 2203, 2206, 2216, 2220, 2222, 2226, 2236, 2241, 2246, 2256, 2260, 2266, 2276, 2286, 2296, 2301, 2307, 2313, 2317, 2320, 2327, 2332, 2337, 2347, 2351, 2357, 2367, 2370, 2377, 2387, 2397, 2401, 2404, 2408, 2418, 2420, 2423, 2428, 2438, 2442, 2448, 2458, 2461, 2468, 2478, 2480, 2488, 2498, 2501, 2509, 2514, 2519, 2520, 2529, 2533, 2539, 2549, 2552, 2559, 2569, 2571, 2579, 2589, 2590, 2599, 2601, 2605, 2620, 2624, 2643, 2662, 2681, 2701, 2715, 2720, 2734, 2753, 2772, 2791, 2801, 2806, 2820, 2825, 2844, 2863, 2882, 2901, 2916, 2920, 2935, 2954, 2973, 2992, 3000, 3001, 3003, 3006, 3009, 3011, 3016, 3022, 3026, 3030, 3036, 3041, 3046, 3056, 3060, 3066, 3076, 3086, 3096, 3107, 3110, 3111, 3113, 3117, 3127, 3130, 3132, 3137, 3147, 3151, 3157, 3167, 3170, 3177, 3187, 3197, 3204, 3208, 3211, 3218, 3223, 3228, 3230, 3238, 3242, 3248, 3258, 3261, 3268, 3278, 3280, 3288, 3298, 3300, 3309, 3311, 3314, 3319, 3329, 3330, 3333, 3339, 3349, 3352, 3359, 3369, 3371, 3379, 3389, 3390, 3399, 3405, 3411, 3424, 3430, 3443, 3462, 3481, 3511, 3515, 3530, 3534, 3553, 3572, 3591, 3606, 3611, 3625, 3630, 3644, 3663, 3682, 3711, 3716, 3730, 3735, 3754, 3773, 3792, 3807, 3811, 3826, 3830, 3845, 3864, 3883, 3911, 3917, 3930, 3936, 3955, 3974, 3993, 4000, 4002, 4004, 4008, 4018, 4020, 4021, 4023, 4028, 4038, 4040, 4042, 4048, 4058, 4061, 4068, 4078, 4080, 4088, 4098, 4101, 4102, 4109, 4114, 4119, 4121, 4129, 4133, 4139, 4140, 4149, 4152, 4159, 4169, 4171, 4179, 4189, 4190, 4199, 4202, 4205, 4210, 4221, 4224, 4240, 4243, 4262, 4281, 4302, 4315, 4321, 4334, 4340, 4353, 4372, 4391, 4400, 4402, 4406, 4421, 4425, 4440, 4444, 4463, 4482, 4502, 4516, 4521, 4535, 4540, 4554, 4573, 4592, 4602, 4607, 4621, 4626, 4640, 4645, 4664, 4683, 4702, 4717, 4721, 4736, 4740, 4755, 4774, 4793, 4802, 4808, 4821, 4827, 4840, 4846, 4865, 4884, 4902, 4918, 4921, 4937, 4940, 4956, 4975, 4994, 5000, 5005, 5011, 5012, 5024, 5031, 5043, 5050, 5062, 5081, 5112, 5115, 5120, 5131, 5134, 5150, 5153, 5172, 5191, 5201, 5206, 5212, 5225, 5231, 5244, 5250, 5263, 5282, 5310, 5312, 5316, 5331, 5335, 5350, 5354, 5373, 5392, 5407, 5412, 5426, 5431, 5445, 5450, 5464, 5483, 5500, 5512, 5517, 5531, 5536, 5550, 5555, 5574, 5593, 5608, 5612, 5627, 5631, 5646, 5650, 5665, 5684, 5712, 5718, 5731, 5737, 5750, 5756, 5775, 5794, 5809, 5812, 5828, 5831, 5847, 5850, 5866, 5885, 5912, 5919, 5931, 5938, 5950, 5957, 5976, 5995, 6000, 6002, 6003, 6006, 6022, 6025, 6030, 6041, 6044, 6060, 6063, 6082, 6103, 6111, 6116, 6122, 6135, 6141, 6154, 6160, 6173, 6192, 6203, 6207, 6220, 6222, 6226, 6241, 6245, 6260, 6264, 6283, 6301, 6303, 6317, 6322, 6336, 6341, 6355, 6360, 6374, 6393, 6403, 6408, 6410, 6422, 6427, 6441, 6446, 6460, 6465, 6484, 6503, 6518, 6522, 6537, 6541, 6556, 6560, 6575, 6594, 6600, 6603, 6609, 6622, 6628, 6641, 6647, 6660, 6666, 6685, 6703, 6719, 6722, 6738, 6741, 6757, 6760, 6776, 6795, 6803, 6822, 6829, 6841, 6848, 6860, 6867, 6886, 6903, 6922, 6939, 6941, 6958, 6960, 6977, 6996, 7000, 7007, 7013, 7021, 7026, 7032, 7045, 7051, 7064, 7070, 7083, 7102, 7113, 7117, 7130, 7132, 7136, 7151, 7155, 7170, 7174, 7193, 7208, 7211, 7213, 7227, 7232, 7246, 7251, 7265, 7270, 7284, 7313, 7318, 7320, 7332, 7337, 7351, 7356, 7370, 7375, 7394, 7401, 7409, 7413, 7428, 7432, 7447, 7451, 7466, 7470, 7485, 7510, 7513, 7519, 7532, 7538, 7551, 7557, 7570, 7576, 7595, 7613, 7629, 7632, 7648, 7651, 7667, 7670, 7686, 7700, 7713, 7732, 7739, 7751, 7758, 7770, 7777, 7796, 7813, 7832, 7849, 7851, 7868, 7870, 7887, 7913, 7932, 7951, 7959, 7970, 7978, 7997, 8000, 8004, 8008, 8012, 8023, 8027, 8040, 8042, 8046, 8061, 8065, 8080, 8084, 8104, 8118, 8121, 8123, 8137, 8142, 8156, 8161, 8175, 8180, 8194, 8202, 8204, 8209, 8223, 8228, 8230, 8242, 8247, 8261, 8266, 8280, 8285, 8304, 8311, 8319, 8323, 8338, 8342, 8357, 8361, 8376, 8380, 8395, 8404, 8420, 8423, 8429, 8442, 8448, 8461, 8467, 8480, 8486, 8501, 8504, 8523, 8539, 8542, 8558, 8561, 8577, 8580, 8596, 8604, 8610, 8623, 8642, 8649, 8661, 8668, 8680, 8687, 8704, 8723, 8742, 8759, 8761, 8778, 8780, 8797, 8800, 8804, 8823, 8842, 8861, 8869, 8880, 8888, 8904, 8923, 8942, 8961, 8979, 8980, 8998, 9000, 9003, 9009, 9014, 9028, 9031, 9033, 9047, 9052, 9066, 9071, 9085, 9090, 9112, 9114, 9119, 9133, 9138, 9140, 9152, 9157, 9171, 9176, 9190, 9195, 9214, 9221, 9229, 9233, 9248, 9252, 9267, 9271, 9286, 9290, 9302, 9314, 9330, 9333, 9339, 9352, 9358, 9371, 9377, 9390, 9396, 9411, 9414, 9433, 9449, 9452, 9468, 9471, 9487, 9490, 9514, 9520, 9533, 9552, 9559, 9571, 9578, 9590, 9597, 9601, 9614, 9633, 9652, 9669, 9671, 9688, 9690, 9710, 9714, 9733, 9752, 9771, 9779, 9790, 9798, 9814, 9833, 9852, 9871, 9889, 9890, 9900, 9914, 9933, 9952, 9971, 9990, 9999, 10000, 10001, 10003, 10004, 10010, 10011, 10013, 10015, 10020, 10023, 10026, 10030, 10033, 10037, 10043, 10048, 10053, 10059, 10063, 10073, 10083, 10093, 10100, 10101, 10102, 10104, 10106, 10114, 10117, 10120, 10121, 10124, 10128, 10134, 10139, 10140, 10144, 10154, 10164, 10174, 10184, 10194, 10200, 10201, 10205, 10208, 10212, 10215, 10219, 10220, 10225, 10231, 10235, 10245, 10250, 10255, 10265, 10275, 10285, 10295, 10301, 10303, 10306, 10316, 10320, 10322, 10326, 10336, 10341, 10346, 10356, 10360, 10366, 10376, 10386, 10396, 10401, 10407, 10413, 10417, 10420, 10427, 10432, 10437, 10447, 10451, 10457, 10467, 10470, 10477, 10487, 10497, 10501, 10504, 10508, 10518, 10520, 10523, 10528, 10538, 10542, 10548, 10558, 10561, 10568, 10578, 10580, 10588, 10598, 10601, 10609, 10614, 10619, 10620, 10629, 10633, 10639, 10649, 10652, 10659, 10669, 10671, 10679, 10689, 10690, 10699, 10701, 10705, 10720, 10724, 10743, 10762, 10781, 10801, 10815, 10820, 10834, 10853, 10872, 10891, 10901, 10906, 10920, 10925, 10944, 10963, 10982, 11000, 11001, 11005, 11007, 11011, 11012, 11015, 11018, 11025, 11029, 11030, 11031, 11035, 11045, 11050, 11055, 11065, 11075, 11085, 11095, 11100, 11103, 11106, 11109, 11110, 11111, 11116, 11122, 11126, 11130, 11136, 11141, 11146, 11156, 11160, 11166, 11176, 11186, 11196, 11207, 11211, 11213, 11217, 11227, 11230, 11232, 11237, 11247, 11251, 11257, 11267, 11270, 11277, 11287, 11297, 11300, 11304, 11308, 11311, 11318, 11323, 11328, 11330, 11338, 11342, 11348, 11358, 11361, 11368, 11378, 11380, 11388, 11398, 11409, 11411, 11414, 11419, 11429, 11430, 11433, 11439, 11449, 11452, 11459, 11469, 11471, 11479, 11489, 11490, 11499, 11505, 11511, 11524, 11530, 11543, 11562, 11581, 11611, 11615, 11630, 11634, 11653, 11672, 11691, 11706, 11711, 11725, 11730, 11744, 11763, 11782, 11811, 11816, 11830, 11835, 11854, 11873, 11892, 11907, 11911, 11926, 11930, 11945, 11964, 11983, 12002, 12007, 12013, 12017, 12020, 12021, 12027, 12032, 12037, 12040, 12047, 12051, 12057, 12067, 12070, 12077, 12087, 12097, 12100, 12101, 12102, 12104, 12108, 12118, 12121, 12123, 12128, 12138, 12140, 12142, 12148, 12158, 12161, 12168, 12178, 12180, 12188, 12198, 12202, 12209, 12210, 12214, 12219, 12221, 12229, 12233, 12239, 12240, 12249, 12252, 12259, 12269, 12271, 12279, 12289, 12290, 12299, 12302, 12305, 12321, 12324, 12340, 12343, 12362, 12381, 12400, 12402, 12415, 12421, 12434, 12440, 12453, 12472, 12491, 12502, 12506, 12521, 12525, 12540, 12544, 12563, 12582, 12602, 12616, 12621, 12635, 12640, 12654, 12673, 12692, 12702, 12707, 12721, 12726, 12740, 12745, 12764, 12783, 12802, 12817, 12821, 12836, 12840, 12855, 12874, 12893, 12902, 12908, 12921, 12927, 12940, 12946, 12965, 12984, 13009, 13011, 13012, 13014, 13019, 13029, 13031, 13033, 13039, 13049, 13050, 13052, 13059, 13069, 13071, 13079, 13089, 13090, 13099, 13100, 13105, 13112, 13120, 13124, 13131, 13143, 13150, 13162, 13181, 13201, 13212, 13215, 13231, 13234, 13250, 13253, 13272, 13291, 13306, 13310, 13312, 13325, 13331, 13344, 13350, 13363, 13382, 13412, 13416, 13431, 13435, 13450, 13454, 13473, 13492, 13500, 13507, 13512, 13526, 13531, 13545, 13550, 13564, 13583, 13612, 13617, 13631, 13636, 13650, 13655, 13674, 13693, 13708, 13712, 13727, 13731, 13746, 13750, 13765, 13784, 13812, 13818, 13831, 13837, 13850, 13856, 13875, 13894, 13909, 13912, 13928, 13931, 13947, 13950, 13966, 13985, 14002, 14003, 14015, 14022, 14030, 14034, 14041, 14053, 14060, 14072, 14091, 14100, 14103, 14106, 14111, 14122, 14125, 14141, 14144, 14160, 14163, 14182, 14203, 14216, 14220, 14222, 14235, 14241, 14254, 14260, 14273, 14292, 14301, 14303, 14307, 14322, 14326, 14341, 14345, 14360, 14364, 14383, 14403, 14410, 14417, 14422, 14436, 14441, 14455, 14460, 14474, 14493, 14503, 14508, 14522, 14527, 14541, 14546, 14560, 14565, 14584, 14600, 14603, 14618, 14622, 14637, 14641, 14656, 14660, 14675, 14694, 14703, 14709, 14722, 14728, 14741, 14747, 14760, 14766, 14785, 14803, 14819, 14822, 14838, 14841, 14857, 14860, 14876, 14895, 14903, 14922, 14929, 14941, 14948, 14960, 14967, 14986, 15013, 15016, 15021, 15032, 15035, 15051, 15054, 15070, 15073, 15092, 15100, 15102, 15107, 15113, 15126, 15130, 15132, 15145, 15151, 15164, 15170, 15183, 15211, 15213, 15217, 15232, 15236, 15251, 15255, 15270, 15274, 15293, 15308, 15313, 15320, 15327, 15332, 15346, 15351, 15365, 15370, 15384, 15401, 15413, 15418, 15432, 15437, 15451, 15456, 15470, 15475, 15494, 15509, 15510, 15513, 15528, 15532, 15547, 15551, 15566, 15570, 15585, 15613, 15619, 15632, 15638, 15651, 15657, 15670, 15676, 15695, 15700, 15713, 15729, 15732, 15748, 15751, 15767, 15770, 15786, 15813, 15832, 15839, 15851, 15858, 15870, 15877, 15896, 15913, 15932, 15949, 15951, 15968, 15970, 15987, 16004, 16012, 16017, 16023, 16036, 16040, 16042, 16055, 16061, 16074, 16080, 16093, 16100, 16104, 16108, 16121, 16123, 16127, 16142, 16146, 16161, 16165, 16180, 16184, 16202, 16204, 16218, 16223, 16230, 16237, 16242, 16256, 16261, 16275, 16280, 16294, 16304, 16309, 16311, 16323, 16328, 16342, 16347, 16361, 16366, 16380, 16385, 16404, 16419, 16420, 16423, 16438, 16442, 16457, 16461, 16476, 16480, 16495, 16501, 16504, 16523, 16529, 16542, 16548, 16561, 16567, 16580, 16586, 16604, 16610, 16623, 16639, 16642, 16658, 16661, 16677, 16680, 16696, 16704, 16723, 16742, 16749, 16761, 16768, 16780, 16787, 16800, 16804, 16823, 16842, 16859, 16861, 16878, 16880, 16897, 16904, 16923, 16942, 16961, 16969, 16980, 16988, 17003, 17014, 17018, 17031, 17033, 17037, 17052, 17056, 17071, 17075, 17090, 17094, 17100, 17109, 17112, 17114, 17128, 17133, 17140, 17147, 17152, 17166, 17171, 17185, 17190, 17214, 17219, 17221, 17233, 17238, 17252, 17257, 17271, 17276, 17290, 17295, 17302, 17314, 17329, 17330, 17333, 17348, 17352, 17367, 17371, 17386, 17390, 17411, 17414, 17433, 17439, 17452, 17458, 17471, 17477, 17490, 17496, 17514, 17520, 17533, 17549, 17552, 17568, 17571, 17587, 17590, 17601, 17614, 17633, 17652, 17659, 17671, 17678, 17690, 17697, 17710, 17714, 17733, 17752, 17769, 17771, 17788, 17790, 17814, 17833, 17852, 17871, 17879, 17890, 17898, 17900, 17914, 17933, 17952, 17971, 17989, 17990, 18005, 18019, 18022, 18024, 18038, 18043, 18050, 18057, 18062, 18076, 18081, 18095, 18100, 18103, 18105, 18124, 18129, 18131, 18143, 18148, 18162, 18167, 18181, 18186, 18205, 18212, 18224, 18239, 18240, 18243, 18258, 18262, 18277, 18281, 18296, 18305, 18321, 18324, 18343, 18349, 18362, 18368, 18381, 18387, 18402, 18405, 18424, 18430, 18443, 18459, 18462, 18478, 18481, 18497, 18505, 18511, 18524, 18543, 18562, 18569, 18581, 18588, 18605, 18620, 18624, 18643, 18662, 18679, 18681, 18698, 18701, 18705, 18724, 18743, 18762, 18781, 18789, 18805, 18810, 18824, 18843, 18862, 18881, 18899, 18905, 18924, 18943, 18962, 18981, 19013, 19015, 19034, 19039, 19041, 19053, 19058, 19072, 19077, 19091, 19096, 19100, 19115, 19122, 19134, 19149, 19150, 19153, 19168, 19172, 19187, 19191, 19203, 19215, 19231, 19234, 19253, 19259, 19272, 19278, 19291, 19297, 19312, 19315, 19334, 19340, 19353, 19369, 19372, 19388, 19391, 19415, 19421, 19434, 19453, 19472, 19479, 19491, 19498, 19502, 19515, 19530, 19534, 19553, 19572, 19589, 19591, 19611, 19615, 19634, 19653, 19672, 19691, 19699, 19715, 19720, 19734, 19753, 19772, 19791, 19801, 19815, 19834, 19853, 19872, 19891, 19910, 19915, 19934, 19953, 19972, 19991, 20000, 20002, 20003, 20006, 20008, 20010, 20016, 20019, 20020, 20021, 20022, 20026, 20036, 20040, 20041, 20046, 20056, 20060, 20066, 20076, 20086, 20096, 20100, 20101, 20102, 20107, 20113, 20117, 20121, 20127, 20132, 20137, 20140, 20147, 20151, 20157, 20167, 20170,

Many thanks again, Giorgos and Hans! The seq was accepted by Neil and the OEIS a few hours ago (thus the opening and closing blades — see here).