Problems & Puzzles: Puzzles

 Problem 79.  More on Egyptian fractions-II Egyptian fractions is an issue that we have posted several times, in Problem 35 and in the Puzzles 897, 899 and 1020. Here we go again exploring an new feature of the same issue, as proposed by Emmanuel. Emmanuel Vantieghem sent the following nice and new problem: For the puzzle 1020, Emmanuel Vantieghem wrote on November 26, 2020: With some refinements of my method I succeeded in improving my previous record. You will find  54642  squarefree numbers (each with three distinct prime factors) whose reciprocals sum to 1 in the annex. Now himself proposes the following question: If we drop the condition of square freeness, fewer numbers are sufficient... my best solution is one with 914 terms: Here is my result : S = 12, 18, 20, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, \ 98, 99, 102, 105, 110, 114, 116, 117, 124, 130, 138, 147, 148, 153, \ 154, 164, 165, 170, 174, 175, 182, 186, 188, 190, 195, 207, 222, 230, \ 231, 236, 238, 242, 245, 246, 255, 258, 261, 266, 273, 275, 279, 282, \ 284, 285, 286, 290, 310, 318, 322, 325, 333, 338, 345, 354, 357, 363, \ 366, 369, 370, 374, 385, 387, 399, 402, 406, 410, 418, 426, 429, 430, \ 434, 435, 442, 455, 465, 470, 475, 477, 483, 506, 530, 531, 539, 549, \ 561, 574, 578, 595, 603, 605, 609, 610, 615, 627, 637, 638, 639, 645, \ 646, 651, 658, 663, 670, 682, 705, 715, 722, 741, 742, 754, 777, 782, \ 795, 826, 833, 845, 847, 861, 867, 897, 902, 903, 925, 931, 935, 938, \ 946, 957, 962, 969, 986, 1001, 1015, 1023, 1025, 1045, 1054, 1066, \ 1075, 1083, 1085, 1102, 1105, 1113, 1118, 1131, 1166, 1173, 1175, \ 1178, 1209, 1221, 1235, 1258, 1281, 1298, 1309, 1311, 1334, 1353, \ 1378, 1394, 1406, 1407, 1421, 1426, 1445, 1462, 1463, 1474, 1475, \ 1479, 1519, 1525, 1551, 1558, 1562, 1581, 1587, 1598, 1599, 1645, \ 1682, 1702, 1705, 1742, 1767, 1771, 1775, 1798, 1802, 1813, 1885, \ 1886, 1887, 1947, 1978, 2006, 2013, 2023, 2035, 2057, 2067, 2074, \ 2091, 2093, 2109, 2146, 2185, 2255, 2261, 2278, 2299, 2301, 2337, \ 2378, 2387, 2414, 2438, 2465, 2523, 2527, 2542, 2546, 2597, 2635, \ 2645, 2666, 2697, 2698, 2717, 2726, 2738, 2769, 2783, 2795, 2806, \ 2849, 2873, 2883, 2891, 2989, 3021, 3034, 3059, 3074, 3082, 3111, \ 3243, 3286, 3311, 3335, 3355, 3362, 3363, 3422, 3441, 3445, 3451, \ 3477, 3478, 3479, 3509, 3526, 3538, 3565, 3685, 3703, 3741, 3751, \ 3757, 3854, 3857, 3886, 3887, 3922, 3971, 3999, 4042, 4071, 4081, \ 4089, 4118, 4154, 4205, 4277, 4301, 4346, 4366, 4402, 4403, 4418, \ 4477, 4514, 4551, 4558, 4611, 4669, 4693, 4773, 4805, 4838, 4879, \ 4899, 4901, 4921, 4929, 4945, 4958, 4961, 4982, 5002, 5015, 5043, \ 5074, 5117, 5133, 5159, 5185, 5203, 5246, 5307, 5365, 5491, 5494, \ 5546, 5547, 5593, 5618, 5681, 5687, 5734, 5762, 5781, 5819, 5822, \ 5887, 6063, 6106, 6177, 6231, 6235, 6251, 6254, 6298, 6307, 6355, \ 6365, 6466, 6519, 6601, 6627, 6727, 6745, 6815, 6837, 6845, 6851, \ 6877, 6962, 7015, 7049, 7102, 7139, 7257, 7259, 7267, 7285, 7337, \ 7381, 7503, 7511, 7526, 7567, 7579, 7585, 7611, 7657, 7869, 7881, \ 7906, 7973, 8041, 8113, 8241, 8303, 8378, 8381, 8405, 8427, 8437, \ 8569, 8591, 8601, 8643, 8662, 8671, 8733, 8845, 8957, 8959, 8978, \ 9061, 9159, 9245, 9269, 9361, 9455, 9499, 9514, 9699, 9805, 9889, \ 9971, 10011, 10051, 10082, 10105, 10309, 10443, 10571, 10653, 10693, \ 10797, 10879, 10915, 10933, 11033, 11045, 11137, 11191, 11767, 11849, \ 11859, 11951, 11977, 11999, 12095, 12121, 12427, 12455, 12493, 12505, \ 12617, 12673, 12803, 12943, 13091, 13135, 13277, 13467, 13489, 13583, \ 13735, 13949, 13981, 14003, 14045, 14147, 14297, 14335, 14413, 14555, \ 14573, 14663, 14801, 14807, 14839, 14911, 14927, 15059, 15123, 15265, \ 15281, 15317, 15407, 15463, 15523, 15635, 15691, 15799, 15979, 16165, \ 16169, 16211, 16337, 17051, 17119, 17405, 17501, 17629, 17641, 17719, \ 17797, 17963, 18073, 18239, 18259, 18361, 18491, 18605, 18815, 18821, \ 18941, 19343, 19363, 19393, 19499, 19573, 19663, 19721, 20119, 20339, \ 20435, 20519, 20683, 20945, 21197, 21229, 21371, 21373, 21655, 21689, \ 21793, 21853, 21889, 22043, 22103, 22231, 22445, 22649, 22747, 22847, \ 22919, 23161, 23273, 23851, 23903, 24037, 24187, 24211, 24299, 24367, \ 24679, 24863, 25051, 25205, 25259, 25783, 25789, 26011, 26047, 26129, \ 26341, 26381, 27001, 27047, 27269, 27401, 27671, 27761, 28037, 28249, \ 28577, 28609, 28613, 28681, 28717, 28823, 28853, 28897, 29279, 29323, \ 29563, 29627, 30073, 30217, 30229, 30659, 31117, 31487, 31939, 32021, \ 32513, 32759, 33263, 33337, 33511, 33583, 34357, 34397, 34481, 34639, \ 34751, 34891, 35003, 35131, 35287, 35351, 35563, 35711, 36517, 36707, \ 36859, 37259, 37417, 37553, 37559, 37789, 38291, 38399, 38663, 39121, \ 39353, 39401, 39463, 39527, 39589, 39689, 39701, 40549, 40651, 40687, \ 40931, 40937, 41123, 41323, 41819, 41971, 42029, 42143, 42253, 42347, \ 42439, 42517, 42527, 43129, 43483, 43993, 44321, 44573, 44689, 44957, \ 45103, 45167, 46079, 46139, 46483, 47141, 47357, 47519, 47641, 47771, \ 48203, 48739, 48749, 48977, 49321, 49379, 49619, 49913, 50209, 50431, \ 50623, 50807, 50933, 51301, 51389, 51911, 52111, 52327, 52417, 53041, \ 53159, 53371, 53533, 54473, 54653, 54739, 54839, 54961, 55309, 55637, \ 56129, 56303, 56347, 56699, 56869, 57293, 57319, 58351, 58357, 58609, \ 58621, 59177, 59413, 59711, 60421, 61183, 61427, 61841, 62197, 63017, \ 63181, 63257, 63307, 63779, 63829, 64061, 64343, 64387, 64607, 65231, \ 65453, 65533, 65941, 66139, 66263, 67363, 67469, 67673, 68231, 69479, \ 69967, 70219, 70699, 71213, 71299, 71891, 71921, 72239, 72283, 72529, \ 72557, 73573, 73627, 75809, 76313, 76751, 76849, 77221, 77653, 78647, \ 79007, 80063, 80401, 80417, 80771, 80869, 81313, 81437, 81461, 81673, \ 81733, 82861, 83509, 83549, 84419, 85291, 85583, 85697, 85963, 86549, \ 86903, 87079, 88537, 88877, 89093, 89311, 90241, 90383, 91321, 91723, \ 92167, 92537, 93757, 93869, 94643, 94987, 95779, 96347, 96937, 97051, \ 97619, 97997, 99179, 99613, 100949, 102131, 102541, 102601, 102979, \ 103247, 103447, 103933, 104017, 107113, 107909, 107911, 109091, \ 109127, 109411, 110081, 112627, 112789, 112961, 114637, 115169, \ 115351, 115699, 115943, 116513, 116653, 117547, 118523, 119351, \ 119621, 121481, 122543, 123281, 123469, 123883, 125173, 128797, \ 130181, 130331, 131279, 131387, 132553, 133163, 134261, 134461, \ 134749, 135407, 136817, 137953, 139159, 139231, 142721, 143491, \ 146189, 146261, 147467, 148003, 149683, 151219, 151951, 154283, \ 154757, 154993, 156271, 156839, 161809, 162073, 163607, 166093, \ 167567, 169153, 169979, 171749, 174887, 176009, 176861, 177571, \ 184049, 184493, 185791, 186233, 186517, 188203, 190747, 192089, \ 193027, 196883, 199439, 203557, 204551, 206681, 209509, 210983, \ 216611, 216763, 222017, 223579, 229543, 233227, 236927, 237917, \ 247151, 249307, 252121, 255529, 264191, 264851, 273829, 280663, 290177}. There are  914 elements in  S.    Each element of  S  has three prime factors (as 12, 18, 20, 28  and 284 more terms ) and their reciprocals sum to 1. It is 'minimal' with respect to the following two operations: 1) Take two elements  a, b  of  S.  If the sum  1/a + 1/b  can be written as  1/c  (with  c  a natural number), then :      either  c  has not three prime factors      or  c  is an element of  S (so, you cannot replace  a, b  by  c). 2) Take three elements  a, b, c  of  S.  If the sum  1/a + 1/b + 1/c  can be written as  1/d  with  d  a naturel number, then :      either  d  has not three prime factors      or  d  is an element of  S (so, you cannot replace  a, b, c  by  d. But, there are other operations possible that could replace  k  elements of  S  by a set of  n  elements with three prime factors that are not in  S  where  n < k. This is however difficult to treat because there are many possibilities. Q. Can you get a solution with less than 914 terms?

Emmanuel Vantieghem wrote on December 18, 2020:

This is a solution with  688  numbers :

{12, 18, 20, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 174, 175, 182, 186, 190, 195, 207, 212, 222, 230, 231, 238, 242, 245, 246, 255, 258, 261, 266, 273, 275, 279, 282, 284, 285, 286, 290, 310, 318, 322, 325, 333, 338, 345, 354, 357, 363, 366, 369, 370, 374, 385, 387, 399, 402, 406, 410, 418, 426, 429, 430, 434, 442, 455, 465, 470, 475, 477, 483, 494, 506, 518, 530, 531, 539, 555, 561, 574, 578, 590, 595, 598, 603, 605, 609, 610, 615, 627, 637, 638, 639, 645, 646, 651, 658, 663, 665, 682, 705, 710, 715, 722, 741, 742, 754, 759, 777, 782, 795, 814, 826, 833, 845, 847, 861, 867, 874, 885, 897, 902, 903, 915, 925, 931, 935, 938, 946, 957, 962, 969, 986, 987, 1001, 1015, 1025, 1034, 1054, 1058, 1066, 1075, 1083, 1102, 1105, 1113, 1131, 1166, 1173, 1175, 1178, 1209, 1221, 1222, 1235, 1258, 1298, 1309, 1334, 1353, 1394, 1407, 1419, 1421, 1445, 1475, 1479, 1519, 1525, 1534, 1551, 1558, 1581, 1586, 1587, 1598, 1599, 1634, 1645, 1702, 1705, 1742, 1767, 1771, 1775, 1813, 1885, 1887, 1947, 1978, 2006, 2023, 2035, 2057, 2067, 2074, 2091, 2093, 2109, 2139, 2146, 2261, 2278, 2299, 2337, 2343, 2387, 2414, 2431, 2451, 2465, 2523, 2527, 2546, 2597, 2635, 2645, 2703, 2717, 2726, 2738, 2769, 2783, 2795, 2873, 2883, 2891, 2989, 3034, 3059, 3074, 3082, 3182, 3266, 3286, 3311, 3422, 3445, 3451, 3477, 3478, 3479, 3485, 3509, 3565, 3621, 3703, 3741, 3751, 3757, 3854, 3886, 3887, 3922, 3971, 4042, 4071, 4081, 4154, 4205, 4277, 4301, 4366, 4403, 4418, 4477, 4551, 4558, 4669, 4693, 4697, 4805, 4838, 4899, 4901, 4921, 4929, 4958, 4961, 4982, 5015, 5159, 5203, 5307, 5365, 5491, 5494, 5547, 5593, 5618, 5681, 5687, 5762, 5797, 5819, 5822, 5829, 5887, 6063, 6177, 6298, 6307, 6355, 6365, 6519, 6627, 6727, 6815, 6845, 6877, 6962, 7049, 7102, 7139, 7257, 7267, 7337, 7503, 7567, 7585, 7973, 8041, 8303, 8378, 8381, 8405, 8427, 8437, 8601, 8643, 8671, 8733, 8845, 8957, 8978, 9245, 9361, 9455, 9499, 9514, 9699, 9805, 9889, 9971, 10051, 10082, 10105, 10309, 10571, 10693, 10797, 10933, 11033, 11045, 11137, 11191, 11339, 11407, 11977, 11999, 12095, 12121, 12427, 12455, 12493, 12505, 12803, 12943, 13091, 13583, 13949, 13981, 14003, 14045, 14335, 14413, 14801, 14807, 14839, 15059, 15281, 15317, 15463, 15523, 15979, 16337, 17051, 17405, 17641, 17719, 17797, 17963, 18491, 18605, 18815, 18821, 18941, 19343, 19363, 19393, 19573, 19663, 19721, 20339, 20435, 20683, 21197, 21229, 21373, 21689, 21853, 22043, 22103, 22445, 22649, 22747, 22919, 23161, 23273, 23851, 23903, 24013, 24037, 24187, 24299, 24367, 24863, 25789, 26011, 26047, 26341, 27047, 27671, 28037, 28249, 28577, 28613, 28717, 28823, 28897, 29279, 29563, 29627, 30217, 30659, 31117, 31487, 31939, 32021, 33263, 33583, 34397, 34751, 34891, 35003, 35131, 35287, 35563, 35711, 36517, 37553, 38291, 38663, 39121, 39353, 39401, 39463, 39527, 39589, 39701, 40931, 41123, 41819, 41971, 42029, 42143, 42253, 42439, 42527, 43483, 43993, 44573, 44957, 45103, 45167, 46139, 47357, 47519, 47771, 48203, 48749, 48977, 49321, 49379, 49913, 50209, 50431, 50807, 50933, 51301, 51911, 52111, 52417, 53041, 53159, 53371, 54473, 54653, 54839, 55637, 56129, 56303, 56699, 56869, 57319, 58357, 58609, 58621, 59177, 59711, 61183, 61427, 61841, 62197, 63257, 63307, 63779, 64061, 64343, 64387, 64607, 65231, 65453, 65533, 65941, 67469, 67673, 68231, 70219, 70699, 71213, 71891, 72283, 72557, 73627, 75809, 76751, 76849, 77221, 77653, 78647, 79007, 80063, 80401, 80417, 80771, 80869, 81313, 81437, 81461, 81673, 81733, 83509, 85291, 85583, 86903, 87079, 88877, 89093, 90241, 90383, 91321, 91723, 93869, 94987, 96347, 96937, 97051, 97997, 99179, 99613, 100949, 102131, 102541, 102601, 102979, 103247, 103933, 107113, 107909, 107911, 109091, 109411, 110081, 112627, 112789, 115169, 115351, 115699, 115943, 116513, 117547, 118523, 119351, 119621, 121481, 122543, 123469, 123883, 125173, 128797, 130181, 130331, 131279, 132553, 133163, 134461, 134749, 135407, 136817, 137953, 139159, 139231, 142721, 146189, 148003, 151951, 156271, 156839, 161809, 162073, 163607, 166093, 167567, 169979, 171749, 174887, 176009, 184493, 185791, 186517, 188203, 190747, 193027, 196883, 199439, 203557, 204551, 206681, 210983, 216611, 216763, 229543, 233227, 236927, 237917, 247151, 249307, 264191, 264851, 273829, 280663, 290177}

There are  266  non squarefree elements in it.

I think there might be solutions with less elements than  688.

***

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