In this week’s Fiddler, you’re looking for numbers of riders where two distinct formations between a triangle and a rhombus are possible.
For Extra Credit, find the fewest number of riders needed so that there are three distinct formations between triangle and a rhombus.
(Note: The sequence of numbers for which there are at least three arrangements also appears to be missing from OEIS. The same goes for the sequence that has the smallest number needed for two formations, followed by the smallest number needed for three formations, followed by the smallest number needed for four formations, and so on. Who will be first to submit all these sequences?)
141 is the first number that can be expressed as three formations:
Here is a list of the first 1000 numbers with exactly three distinct pelotons:
141, 190, 253, 288, 435, 463, 484, 631, 778, 831, 841, 925, 1023, 1120, 1170, 1219, 1268, 1366, 1464, 1471, 1611, 1660, 1698, 1709, 1905, 1933, 2050, 2101, 2199, 2248, 2276, 2346, 2565, 2591, 2640, 2689, 2763, 2836, 2854, 2934, 2983, 3081, 3108, 3522, 3669, 3718, 3721, 3865, 3914, 3963, 4166, 4355, 4404, 4411, 4453, 4551, 4588, 4600, 4663, 4695, 4747, 4833, 5041, 5139, 5224, 5286, 5433, 5455, 5461, 5580, 5629, 5646, 5678, 5923, 6021, 6033, 6070, 6315, 6322, 6351, 6364, 6511, 6514, 6658, 6756, 6811, 6900, 6931, 7001, 7050, 7189, 7197, 7478, 7491, 7540, 7568, 7834, 7981, 8056, 8079, 8128, 8233, 8324, 8373, 8398, 8529, 8560, 8569, 8634, 8716, 8814, 8863, 9010, 9108, 9206, 9255, 9283, 9451, 9490, 9501, 9549, 9598, 9696, 9790, 9843, 9876, 9985, 10079, 10186, 10529, 10774, 10872, 10921, 10970, 11043, 11166, 11173, 11215, 11235, 11313, 11411, 11460, 11509, 11572, 11950, 12097, 12244, 12293, 12373, 12489, 12630, 12636, 12783, 12930, 12969, 13126, 13159, 13175, 13238, 13258, 13518, 13714, 13836, 14008, 14113, 14155, 14400, 14493, 14498, 14596, 14694, 14746, 14841, 14919, 14939, 14988, 15256, 15281, 15321, 15331, 15380, 15570, 15576, 15625, 15772, 15804, 15859, 15870, 16213, 16311, 16360, 16423, 16458, 16600, 16605, 16654, 16726, 16948, 17046, 17095, 17144, 17304, 17391, 17396, 17593, 17634, 17683, 17781, 17920, 17928, 17943, 18139, 18171, 18369, 18449, 18516, 18859, 18908, 19104, 19398, 19447, 19534, 19545, 19605, 19790, 19951, 20035, 20061, 20231, 20329, 20476, 20483, 20565, 20574, 20650, 20770, 20772, 20868, 21022, 21113, 21211, 21505, 21603, 21639, 21643, 21799, 21814, 21928, 21946, 22093, 22191, 22240, 22485, 22773, 22779, 22795, 22828, 22944, 22975, 23073, 23084, 23122, 23318, 23373, 23416, 23563, 23661, 23739, 23759, 23808, 23905, 23955, 24004, 24151, 24184, 24200, 24268, 24298, 24403, 24543, 24866, 24984, 25180, 25278, 25326, 25474, 25523, 25572, 25685, 25915, 25974, 26160, 26263, 26405, 26503, 26686, 26788, 26841, 26846, 26895, 27051, 27091, 27130, 27238, 27336, 27679, 27728, 27971, 27973, 27997, 28071, 28120, 28414, 28441, 28500, 28561, 28659, 28710, 28806, 29029, 29100, 29149, 29247, 29296, 29394, 29590, 29616, 29731, 29835, 30031, 30129, 30276, 30309, 30325, 30423, 30472, 30501, 30598, 30616, 30619, 30694, 30766, 30864, 31158, 31176, 31308, 31354, 31593, 31648, 31674, 31697, 31729, 31921, 32040, 32043, 32089, 32203, 32334, 32481, 32554, 32593, 32775, 32910, 33020, 33069, 33118, 33199, 33261, 33265, 33461, 33488, 33510, 33559, 33706, 33804, 33853, 34147, 34245, 34441, 34539, 34588, 34651, 34784, 34833, 34933, 34980, 35068, 35091, 35176, 35225, 35421, 35470, 35511, 35568, 35666, 35764, 35800, 36009, 36156, 36303, 36460, 36597, 36666, 36711, 36772, 36793, 37038, 37234, 37245, 37359, 37493, 37534, 37626, 37675, 37969, 38263, 38361, 38401, 38410, 38508, 38655, 38851, 38949, 38998, 39194, 39268, 39609, 39635, 39698, 39846, 40076, 40125, 40134, 40138, 40272, 40419, 40468, 40909, 40958, 41056, 41110, 41154, 41291, 41399, 41497, 41580, 41595, 41695, 41725, 41791, 41815, 41966, 42036, 42085, 42254, 42330, 42447, 42624, 42673, 42722, 42736, 42771, 42783, 42820, 43114, 43125, 43359, 43496, 43506, 43513, 43653, 43800, 43849, 44086, 44143, 44181, 44388, 44470, 45025, 45368, 45858, 46008, 46113, 46204, 46250, 46299, 46371, 46446, 46495, 46536, 46593, 46789, 46858, 47015, 47071, 47083, 47181, 47295, 47328, 47503, 47524, 47544, 47573, 47622, 47649, 47916, 47938, 48073, 48259, 48322, 48504, 48553, 48651, 48700, 48896, 49094, 49131, 49148, 49239, 49383, 49480, 49533, 49660, 49680, 49780, 49981, 50023, 50170, 50219, 50220, 50415, 50539, 50611, 50660, 50758, 50813, 50821, 50856, 50905, 51101, 51247, 51406, 51591, 51640, 51738, 51774, 51984, 52228, 52273, 52326, 52522, 52620, 52624, 52669, 52735, 52740, 52816, 52963, 53061, 53140, 53306, 53404, 53718, 53796, 53943, 53961, 54090, 54139, 54188, 54286, 54296, 54421, 54433, 54585, 54629, 54825, 54949, 54972, 55119, 55168, 55263, 55278, 55389, 55479, 55618, 55741, 55756, 55903, 56008, 56050, 56197, 56319, 56344, 56589, 56608, 56834, 56897, 56944, 57030, 57066, 57079, 57158, 57186, 57226, 57275, 57520, 57540, 57765, 57793, 57814, 57961, 58053, 58059, 58255, 58500, 58501, 58549, 58598, 58625, 58647, 58653, 58941, 58990, 59284, 59529, 59676, 59823, 59921, 59970, 60076, 60240, 60264, 60423, 60460, 60523, 60558, 60769, 60803, 61048, 61293, 61298, 61489, 61576, 61734, 61881, 61930, 62099, 62175, 62224, 62328, 62378, 62500, 62616, 62763, 62910, 63008, 63106, 63351, 63449, 63544, 63645, 63694, 63943, 64086, 64122, 64233, 64261, 64411, 64674, 64700, 64723, 64772, 64903, 64968, 65001, 65115, 65164, 65311, 65349, 65458, 65530, 65553, 65605, 65856, 65994, 65997, 66144, 66145, 66340, 66438, 66723, 66781, 67222, 67301, 67418, 67590, 67663, 67810, 67859, 67879, 67908, 68104, 68111, 68175, 68349, 68545, 68611, 68704, 68839, 69072, 69084, 69091, 69133, 69280, 69525, 69721, 69819, 69966, 70113, 70260, 70291, 70299, 70456, 70480, 70505, 70554, 70603, 70750, 70897, 71347, 71779, 71828, 71878, 71926, 71955, 72171, 72214, 72318, 72465, 72503, 72514, 72621, 72661, 72710, 72730, 72808, 72906, 72916, 72936, 73053, 73151, 73249, 73347, 73494, 73543, 73659, 73935, 74096, 74229, 74376, 74526, 74572, 74830, 74915, 74964, 75018, 75111, 75356, 75393, 75435, 75581, 75601, 75799, 75895, 76110, 76260, 76336, 76434, 76483, 76581, 76630, 76728, 76760, 77022, 77039, 77071, 77116, 77120, 77416, 77463, 77610, 77697, 77705, 77721, 77806, 78051, 78220, 78226, 78345, 78394, 78443, 78541, 79080, 79129, 79325, 79374, 79423, 79521, 79668, 79728, 79766, 80011, 80026, 80181, 80256, 80353, 80403, 80478, 80550, 80599, 80746, 80871, 80884, 81089, 81400, 81628, 81775, 81873, 81971, 82069, 82461, 82608, 82618, 82853, 83098, 83245, 83294, 83485, 83505, 83546, 83784, 83833, 83980, 84225, 84274, 84519, 84568, 84574, 84598, 84631, 84715, 84930, 85009, 85058, 85156, 85409, 85508, 85548, 85723, 85744, 85875, 85940, 86086, 86161, 86185, 86283, 86370, 86381, 86626, 86664, 86724, 87018, 87219, 87263, 87331, 87531, 87606, 87748, 87753, 88047, 88109, 88243, 88341, 88354, 88390, 88398, 88488, 88586, 88684, 88831, 88883, 88929, 89076, 89125, 89253, 89370, 89664, 89864, 90214, 90393, 90399, 90421, 90448, 90546, 90564, 90710, 90840, 91134, 91183, 91288, 91575, 91673, 91722, 91771, 91866, 92065, 92155, 92245, 92509, 92914, 92947, 93022, 93094, 93486, 93535, 93600, 93633, 93829, 94074, 94096, 94123, 94155, 94221, 94270, 94564, 94758, 94760, 95005, 95103, 95149, 95154, 95250, 95299, 95446, 95593, 95623, 95683, 95838, 95985, 96034, 96201, 96328, 96490, 96573, 96622, 96720, 96916, 96965, 97068, 97270, 97288, 97308, 97504, 97749, 97847, 97935, 97945, 98043, 98224, 98288, 98328, 98386, 98533, 98631, 98680, 98729, 98778, 98863, 99072, 99076, 99121, 99129, 99366, 99373, 99386, 99415, 99915, 99954, 99958, 100493, 100536, 100556, 100585, 100650, 100738, 100785, 100825, 100934, 100983, 101114
There seem to be triple peloton pairs as well!
Here is the first number that can be expressed as exactly $n$ different formations:
| n | Riders | Formations |
|---|---|---|
| $1$ | $1$ | $1^2 - \sum{0}$ |
| $2$ | $15$ | $4^2 - \sum{1}$ $5^2 - \sum{4}$ |
| $3$ | $141$ | $12^2 - \sum{2}$ $13^2 - \sum{7}$ $14^2 - \sum{10}$ |
| $4$ | $610$ | $25^2 - \sum{5}$ $26^2 - \sum{11}$ $29^2 - \sum{21}$ $31^2 - \sum{26}$ |
| $5$ | $6,903$ | $84^2 - \sum{17}$ $87^2 - \sum{36}$ $91^2 - \sum{52}$ $98^2 - \sum{73}$ $117^2 - \sum{116}$ |
| $6$ | $2,395$ | $49^2 - \sum{3}$ $50^2 - \sum{14}$ $55^2 - \sum{35}$ $56^2 - \sum{38}$ $61^2 - \sum{51}$ $65^2 - \sum{60}$ |
| $7$ | $338,241$ | See code |
| $8$ | $10,606$ | See code |
| $9$ | $40,713$ | See code |
| $10$ | $117,349$ | See code |
| $11$ | $>200,170,034$ | See code |
| $12$ | $98,190$ | See code |
| $13$ | $>203,282,766$ | See code |
| $14$ | $5,750,095$ | See code |
| $15$ | $1,994,931$ | See code |
| $16$ | $434,841$ | See code |
| $17$ | $>201,169,641$ | See code |
| $18$ | $1,669,228$ | See code |
| $19$ | $>203,563,683$ | See code |
| $20$ | $4,811,304$ | See code |
There is clearly a pattern from 5, 7, 11, etc. that a prime number of distinct formations requires many more riders than composite numbers of similar size.
The number of riders necessary is larger than listed above for 11, 13, 17, and 19. I am attempting to tweak my code to avoid memory issues.
Update. Number of riders necessary for 11, 13, 17, and 19 formations, if they exist, is at least $450,000,000$.