Problems & Puzzles: Puzzles

Problem 76.  More on Egyptian fractions

Emmanuel Vantieghem sent the following nice problem:

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:

Find ODD semiprimes whose reciprocals sum up to 1 or prove that this is impossible.

Q. Send your best result

 

Emmanuel Vantieghem wrote on November 14, 2020:


With a ultimate effort I finally found a set of  2775  semiprimes whose reciprocals sum to 1 :

I arrived at a point that I started to believe that such a set was impossible !

{15, 21, 35, 39, 51, 55, 57, 65, 69, 77, 87, 91, 95, 111, 115, 119, 123, 129, 133, 141, 145, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299, 301, 303, 305, 309, 319, 321, 323, 327, 329, 335, 339, 341, 355, 365, 371, 377, 381, 391, 393, 395, 403, 407, 411, 413, 417, 427, 437, 445, 447, 451, 453, 469, 471, 473, 481, 485, 489, 497, 501, 505, 511, 515, 517, 519, 527, 535, 537, 543, 545, 551, 553, 559, 565, 573, 579, 581, 583, 591, 597, 623, 629, 633, 635, 649, 655, 667, 669, 671, 679, 681, 685, 687, 689, 695, 697, 699, 703, 707, 713, 717, 721, 737, 745, 749, 755, 763, 767, 771, 781, 785, 789, 791, 793, 799, 803, 807, 813, 815, 817, 831, 835, 843, 849, 851, 865, 869, 871, 879, 889, 895, 899, 901, 905, 913, 917, 921, 923, 933, 939, 943, 949, 951, 955, 959, 965, 979, 985, 989, 993, 995, 1003, 1011, 1027, 1037, 1041, 1043, 1047, 1055, 1057, 1059, 1067, 1073, 1077, 1081, 1099, 1101, 1111, 1115, 1119, 1121, 1133, 1135, 1137, 1139, 1141, 1145, 1149, 1157, 1159, 1165, 1167, 1169, 1177, 1189, 1191, 1195, 1199, 1203, 1205, 1207, 1211, 1219, 1241, 1243, 1247, 1253, 1255, 1261, 1267, 1285, 1313, 1315, 1333, 1337, 1339, 1345, 1349, 1351, 1355, 1357, 1363, 1379, 1385, 1387, 1391, 1393, 1403, 1405, 1415, 1417, 1457, 1465, 1469, 1477, 1507, 1513, 1529, 1535, 1537, 1541, 1555, 1561, 1565, 1577, 1585, 1589, 1603, 1631, 1633, 1639, 1643, 1649, 1651, 1655, 1661, 1679, 1685, 1687, 1691, 1703, 1711, 1727, 1735, 1745, 1751, 1757, 1763, 1765, 1769, 1781, 1793, 1795, 1799, 1817, 1819, 1829, 1835, 1837, 1841, 1843, 1853, 1865, 1883, 1891, 1895, 1897, 1903, 1909, 1915, 1919, 1921, 1937, 1939, 1943, 1945, 1957, 1961, 1963, 1967, 1969, 1981, 1985, 1991, 2005, 2021, 2041, 2047, 2051, 2059, 2071, 2077, 2101, 2119, 2123, 2147, 2149, 2167, 2171, 2173, 2177, 2183, 2189, 2191, 2201, 2219, 2227, 2231, 2249, 2257, 2263, 2291, 2317, 2321, 2323, 2327, 2353, 2359, 2363, 2369, 2413, 2419, 2429, 2443, 2449, 2453, 2461, 2471, 2479, 2483, 2489, 2491, 2497, 2501, 2507, 2509, 2513, 2519, 2533, 2537, 2561, 2563, 2569, 2573, 2587, 2599, 2603, 2611, 2623, 2627, 2629, 2651, 2653, 2669, 2681, 2701, 2723, 2743, 2747, 2759, 2761, 2771, 2773, 2807, 2813, 2827, 2839, 2867, 2869, 2881, 2893, 2899, 2911, 2921, 2923, 2929, 2941, 2951, 2959, 2977, 2981, 2983, 2987, 2993, 3007, 3013, 3029, 3043, 3047, 3077, 3091, 3097, 3103, 3107, 3113, 3127, 3131, 3133, 3139, 3149, 3151, 3161, 3173, 3193, 3197, 3223, 3233, 3247, 3263, 3277, 3281, 3287, 3293, 3317, 3337, 3341, 3349, 3377, 3379, 3383, 3397, 3401, 3419, 3421, 3427, 3431, 3439, 3443, 3473, 3487, 3497, 3503, 3523, 3551, 3587, 3589, 3599, 3601, 3611, 3629, 3641, 3649, 3653, 3667, 3679, 3683, 3707, 3743, 3749, 3763, 3781, 3791, 3799, 3809, 3811, 3817, 3827, 3839, 3841, 3859, 3869, 3883, 3893, 3937, 3949, 3953, 3959, 3961, 3973, 3979, 3991, 4009, 4031, 4033, 4037, 4043, 4061, 4063, 4069, 4087, 4103, 4117, 4121, 4141, 4163, 4169, 4171, 4181, 4183, 4189, 4223, 4237, 4247, 4267, 4279, 4303, 4307, 4309, 4313, 4321, 4331, 4343, 4351, 4367, 4369, 4379, 4381, 4387, 4393, 4411, 4427, 4429, 4439, 4453, 4469, 4471, 4511, 4537, 4541, 4553, 4559, 4573, 4577, 4579, 4589, 4601, 4619, 4633, 4661, 4667, 4681, 4687, 4709, 4717, 4727, 4747, 4757, 4769, 4771, 4777, 4811, 4819, 4843, 4847, 4849, 4853, 4859, 4867, 4883, 4891, 4897, 4927, 4979, 4981, 5017, 5029, 5053, 5057, 5063, 5069, 5111, 5123, 5129, 5141, 5149, 5161, 5177, 5191, 5221, 5249, 5251, 5263, 5267, 5287, 5293, 5311, 5321, 5339, 5353, 5359, 5363, 5371, 5377, 5389, 5429, 5459, 5497, 5513, 5539, 5543, 5549, 5561, 5567, 5587, 5597, 5609, 5611, 5617, 5627, 5633, 5671, 5699, 5713, 5723, 5729, 5767, 5771, 5773, 5777, 5833, 5891, 5893, 5899, 5909, 5911, 5917, 5921, 5933, 5947, 5959, 5963, 5977, 5983, 6001, 6023, 6031, 6049, 6059, 6077, 6103, 6107, 6109, 6119, 6157, 6161, 6169, 6179, 6187, 6191, 6233, 6239, 6283, 6289, 6313, 6319, 6341, 6371, 6401, 6403, 6407, 6431, 6437, 6439, 6443, 6463, 6467, 6493, 6497, 6499, 6509, 6511, 6527, 6583, 6593, 6613, 6631, 6641, 6649, 6667, 6683, 6697, 6707, 6739, 6749, 6751, 6757, 6817, 6821, 6847, 6887, 6893, 6901, 6913, 6931, 6943, 6973, 6989, 7003, 7009, 7031, 7037, 7061, 7081, 7087, 7093, 7097, 7099, 7141, 7153, 7169, 7171, 7181, 7199, 7201, 7223, 7261, 7279, 7289, 7291, 7303, 7313, 7339, 7363, 7367, 7373, 7379, 7387, 7391, 7409, 7421, 7439, 7453, 7471, 7493, 7519, 7543, 7571, 7597, 7613, 7619, 7627, 7661, 7697, 7729, 7739, 7747, 7751, 7781, 7783, 7801, 7807, 7811, 7831, 7849, 7859, 7897, 7913, 7957, 7967, 7979, 7981, 7991, 8003, 8023, 8027, 8033, 8077, 8083, 8119, 8131, 8137, 8149, 8153, 8159, 8201, 8207, 8213, 8249, 8251, 8257, 8299, 8321, 8339, 8357, 8383, 8399, 8401, 8413, 8441, 8453, 8471, 8473, 8479, 8497, 8507, 8509, 8549, 8557, 8579, 8587, 8611, 8621, 8633, 8639, 8651, 8711, 8717, 8773, 8777, 8791, 8809, 8851, 8881, 8903, 8909, 8917, 8947, 8977, 8989, 9017, 9019, 9047, 9071, 9073, 9077, 9083, 9089, 9131, 9143, 9167, 9169, 9179, 9193, 9211, 9259, 9263, 9271, 9287, 9301, 9307, 9313, 9353, 9379, 9389, 9487, 9509, 9517, 9523, 9553, 9563, 9577, 9589, 9593, 9599, 9617, 9641, 9701, 9703, 9727, 9731, 9761, 9773, 9797, 9799, 9827, 9847, 9853, 9869, 9881, 9917, 9943, 9953, 9983, 9991, 10001, 10019, 10027, 10057, 10063, 10117, 10121, 10123, 10187, 10207, 10237, 10249, 10261, 10277, 10291, 10349, 10363, 10379, 10397, 10403, 10411, 10441, 10447, 10471, 10481, 10519, 10537, 10547, 10553, 10561, 10573, 10579, 10643, 10669, 10679, 10721, 10763, 10783, 10793, 10807, 10817, 10823, 10841, 10873, 10877, 10919, 10921, 10943, 10951, 10961, 10991, 11009, 11021, 11023, 11029, 11041, 11107, 11111, 11129, 11147, 11183, 11227, 11233, 11269, 11281, 11303, 11309, 11327, 11357, 11359, 11371, 11387, 11413, 11461, 11507, 11521, 11563, 11567, 11573, 11581, 11591, 11603, 11629, 11639, 11651, 11653, 11729, 11741, 11749, 11771, 11773, 11797, 11819, 11857, 11873, 11899, 11911, 11929, 11993, 12013, 12017, 12031, 12059, 12079, 12083, 12091, 12127, 12137, 12139, 12169, 12191, 12193, 12247, 12283, 12307, 12317, 12349, 12361, 12367, 12371, 12403, 12431, 12449, 12469, 12533, 12587, 12599, 12629, 12643, 12667, 12707, 12709, 12737, 12751, 12773, 12797, 12827, 12833, 12839, 12851, 12871, 12877, 12913, 12931, 13031, 13081, 13157, 13193, 13199, 13207, 13213, 13231, 13283, 13289, 13301, 13303, 13333, 13373, 13393, 13439, 13459, 13493, 13511, 13561, 13571, 13579, 13589, 13603, 13621, 13631, 13667, 13703, 13747, 13771, 13801, 13817, 13837, 13843, 13847, 13861, 13939, 13943, 13969, 13973, 13987, 14017, 14023, 14039, 14089, 14101, 14111, 14129, 14137, 14141, 14171, 14213, 14219, 14227, 14233, 14257, 14279, 14299, 14309, 14317, 14351, 14359, 14363, 14381, 14393, 14429, 14453, 14473, 14507, 14527, 14579, 14617, 14647, 14659, 14681, 14689, 14701, 14711, 14719, 14803, 14809, 14837, 14857, 14863, 14873, 14893, 14899, 14921, 14933, 14941, 14999, 15007, 15023, 15047, 15049, 15089, 15151, 15163, 15179, 15209, 15229, 15247, 15251, 15293, 15311, 15343, 15347, 15397, 15403, 15437, 15481, 15517, 15529, 15539, 15563, 15611, 15677, 15703, 15707, 15721, 15781, 15811, 15833, 15839, 15853, 15857, 15871, 15931, 15943, 15989, 16013, 16019, 16039, 16043, 16117, 16147, 16171, 16199, 16241, 16259, 16271, 16277, 16297, 16309, 16343, 16351, 16403, 16409, 16441, 16459, 16463, 16469, 16483, 16517, 16531, 16543, 16571, 16579, 16589, 16637, 16697, 16717, 16727, 16781, 16789, 16799, 16801, 16817, 16837, 16867, 16873, 16897, 16969, 16999, 17009, 17071, 17111, 17113, 17141, 17177, 17201, 17219, 17243, 17249, 17263, 17287, 17363, 17399, 17441, 17447, 17473, 17513, 17531, 17533, 17543, 17557, 17593, 17617, 17621, 17711, 17741, 17767, 17813, 17819, 17821, 17861, 17869, 17873, 17933, 17947, 18001, 18023, 18079, 18091, 18113, 18157, 18203, 18209, 18247, 18283, 18323, 18349, 18391, 18407, 18419, 18437, 18467, 18497, 18509, 18511, 18527, 18559, 18643, 18659, 18673, 18703, 18709, 18721, 18727, 18761, 18779, 18841, 18847, 18857, 18871, 18881, 18923, 18961, 18971, 19007, 19027, 19039, 19043, 19093, 19099, 19109, 19153, 19199, 19241, 19291, 19303, 19337, 19339, 19367, 19451, 19493, 19511, 19519, 19529, 19549, 19631, 19637, 19667, 19729, 19769, 19781, 19783, 19829, 19837, 19847, 19879, 19883, 19897, 19939, 19951, 20003, 20087, 20093, 20099, 20191, 20203, 20221, 20227, 20291, 20299, 20303, 20381, 20413, 20437, 20453, 20467, 20473, 20497, 20513, 20557, 20567, 20569, 20591, 20617, 20651, 20687, 20701, 20711, 20737, 20803, 20819, 20827, 20833, 20837, 20971, 21037, 21041, 21079, 21167, 21181, 21209, 21239, 21251, 21253, 21271, 21289, 21293, 21311, 21331, 21353, 21389, 21409, 21449, 21473, 21509, 21533, 21583, 21631, 21653, 21691, 21733, 21797, 21809, 21829, 21877, 21883, 21899, 21971, 22007, 22019, 22081, 22177, 22199, 22213, 22223, 22261, 22327, 22331, 22339, 22357, 22361, 22387, 22411, 22487, 22493, 22499, 22507, 22577, 22579, 22597, 22601, 22657, 22663, 22703, 22733, 22753, 22849, 22873, 22879, 22927, 22951, 22969, 22987, 22991, 22999, 23119, 23129, 23141, 23147, 23183, 23213, 23249, 23323, 23363, 23377, 23381, 23383, 23393, 23407, 23423, 23449, 23489, 23501, 23533, 23587, 23651, 23659, 23701, 23707, 23711, 23729, 23843, 23861, 23927, 23941, 24047, 24053, 24119, 24139, 24163, 24253, 24257, 24287, 24289, 24307, 24319, 24341, 24347, 24461, 24503, 24511, 24523, 24569, 24589, 24601, 24613, 24617, 24637, 24653, 24727, 24743, 24779, 24797, 24823, 24881, 24883, 24929, 24931, 24961, 24991, 25009, 25019, 25021, 25043, 25063, 25159, 25187, 25199, 25217, 25273, 25331, 25351, 25393, 25397, 25477, 25481, 25489, 25511, 25573, 25591, 25651, 25661, 25769, 25777, 25787, 25807, 25813, 25853, 25877, 25957, 25979, 26051, 26057, 26063, 26069, 26077, 26093, 26123, 26149, 26167, 26207, 26219, 26269, 26287, 26311, 26329, 26471, 26483, 26549, 26563, 26599, 26623, 26791, 26797, 26857, 26867, 26869, 26909, 26969, 26989, 27007, 27029, 27089, 27161, 27169, 27193, 27221, 27229, 27233, 27257, 27263, 27323, 27359, 27371, 27413, 27451, 27473, 27499, 27571, 27619, 27641, 27667, 27679, 27707, 27857, 27887, 27959, 27971, 27977, 28013, 28103, 28141, 28187, 28199, 28213, 28321, 28361, 28363, 28381, 28397, 28417, 28421, 28471, 28531, 28583, 28667, 28783, 28801, 28829, 28891, 28907, 28943, 28967, 28981, 28993, 28997, 29041, 29083, 29149, 29177, 29213, 29273, 29299, 29321, 29329, 29353, 29459, 29467, 29503, 29539, 29591, 29593, 29639, 29651, 29719, 29737, 29747, 29779, 29797, 29893, 29941, 29993, 29999, 30049, 30067, 30167, 30179, 30193, 30227, 30257, 30281, 30353, 30361, 30397, 30461, 30523, 30607, 30623, 30629, 30731, 30749, 30847, 30883, 30929, 30967, 30997, 31007, 31061, 31133, 31243, 31301, 31309, 31313, 31351, 31363, 31373, 31411, 31417, 31439, 31457, 31459, 31553, 31571, 31613, 31621, 31679, 31753, 31789, 31831, 31877, 31897, 31921, 31937, 31951, 31979, 32017, 32033, 32107, 32111, 32231, 32239, 32287, 32387, 32399, 32437, 32639, 32663, 32689, 32743, 32849, 32881, 32899, 32951, 33017, 33043, 33109, 33197, 33221, 33233, 33277, 33283, 33389, 33401, 33431, 33463, 33491, 33659, 33667, 33731, 33823, 33853, 33899, 33919, 34037, 34081, 34087, 34093, 34117, 34121, 34163, 34189, 34241, 34277, 34387, 34393, 34417, 34427, 34453, 34547, 34553, 34579, 34621, 34691, 34711, 34717, 34823, 34889, 35047, 35143, 35179, 35183, 35209, 35237, 35249, 35263, 35333, 35369, 35417, 35501, 35599, 35621, 35639, 35653, 35657, 35687, 35689, 35723, 35741, 35821, 35909, 35941, 35947, 35953, 36019, 36031, 36059, 36079, 36089, 36181, 36259, 36287, 36349, 36359, 36391, 36503, 36557, 36581, 36733, 36763, 36811, 36853, 36977, 37001, 37067, 37073, 37127, 37129, 37151, 37211, 37327, 37343, 37391, 37399, 37403, 37523, 37627, 37669, 37673, 37733, 37769, 37771, 37801, 37823, 37837, 37901, 37909, 37949, 37979, 38009, 38021, 38041, 38081, 38243, 38279, 38293, 38383, 38407, 38413, 38419, 38477, 38497, 38503, 38509, 38579, 38683, 38771, 38807, 38897, 38911, 38957, 38989, 39037, 39059, 39131, 39187, 39203, 39211, 39269, 39271, 39283, 39289, 39337, 39407, 39437, 39449, 39617, 39713, 39889, 39911, 39913, 39917, 40003, 40067, 40081, 40097, 40141, 40217, 40247, 40259, 40309, 40349, 40379, 40501, 40619, 40633, 40657, 40727, 40741, 40891, 40913, 40921, 40981, 40991, 41003, 41087, 41273, 41291, 41303, 41311, 41347, 41471, 41527, 41567, 41623, 41693, 41707, 41747, 41827, 41869, 41891, 41917, 41989, 42059, 42149, 42167, 42173, 42233, 42401, 42431, 42479, 42547, 42607, 42673, 42733, 42781, 42799, 42827, 42869, 42881, 42907, 42919, 43139, 43229, 43259, 43273, 43279, 43357, 43361, 43423, 43429, 43489, 43507, 43621, 43657, 43709, 43739, 43811, 43847, 43921, 43931, 43957, 44063, 44069, 44117, 44147, 44173, 44197, 44243, 44323, 44377, 44431, 44461, 44503, 44719, 44831, 44861, 44923, 44929, 44969, 45113, 45151, 45173, 45257, 45313, 45347, 45431, 45457, 45499, 45593, 45649, 45719, 45743, 45803, 45901, 46001, 46003, 46009, 46031, 46127, 46129, 46169, 46243, 46259, 46339, 46357, 46367, 46513, 46517, 46537, 46609, 46637, 46843, 46883, 46927, 46961, 47029, 47077, 47083, 47233, 47261, 47263, 47371, 47477, 47539, 47561, 47603, 47759, 47813, 47867, 47897, 47921, 47941, 47959, 48077, 48133, 48151, 48199, 48233, 48361, 48443, 48509, 48511, 48613, 48641, 48689, 48827, 48863, 48931, 48959, 49051, 49067, 49087, 49141, 49163, 49183, 49319, 49403, 49447, 49583, 49601, 49649, 49769, 49901, 49949, 49981, 50041, 50137, 50173, 50213, 50233, 50279, 50299, 50419, 50629, 50657, 50689, 50693, 50759, 50851, 50861, 50887, 50927, 50959, 51013, 51019, 51101, 51143, 51223, 51269, 51379, 51671, 51703, 51761, 51811, 51847, 51917, 51923, 51937, 51959, 51967, 52001, 52007, 52271, 52303, 52337, 52397, 52447, 52471, 52531, 52597, 52681, 52699, 52891, 52907, 52909, 52939, 52961, 52993, 53033, 53111, 53237, 53293, 53303, 53357, 53387, 53461, 53491, 53531, 53671, 53743, 53803, 53929, 53953, 54053, 54071, 54209, 54227, 54233, 54389, 54479, 54569, 54619, 54683, 54707, 54793, 54841, 54931, 54937, 54953, 55123, 55183, 55189, 55277, 55357, 55417, 55421, 55493, 55567, 55577, 55669, 55687, 55739, 55751, 55919, 55963, 55973, 56027, 56153, 56279, 56291, 56317, 56323, 56363, 56471, 56549, 56561, 56653, 56743, 56759, 56887, 56977, 57067, 57181, 57229, 57263, 57377, 57479, 57539, 57599, 57619, 57721, 57833, 57949, 57961, 58283, 58301, 58307, 58339, 58447, 58483, 58517, 58561, 58637, 58649, 58739, 58853, 58951, 59153, 59249, 59251, 59291, 59401, 59503, 59701, 59713, 59749, 59783, 59821, 59881, 59911, 59947, 59953, 59987, 60023, 60031, 60131, 60227, 60323, 60377, 60409, 60433, 60479, 60547, 60551, 60799, 60997, 61063, 61069, 61073, 61093, 61181, 61267, 61279, 61289, 61423, 61517, 61601, 61661, 61771, 61777, 61823, 61889, 61937, 62059, 62107, 62113, 62287, 62291, 62329, 62429, 62449, 62471, 62663, 62677, 62807, 62857, 62879, 62957, 63083, 63109, 63143, 63169, 63187, 63221, 63293, 63383, 63407, 63433, 63491, 63787, 63883, 63893, 63961, 64241, 64261, 64291, 64349, 64367, 64507, 64529, 64541, 64711, 64769, 64777, 64807, 64829, 64963, 64979, 65041, 65207, 65311, 65339, 65363, 65473, 65567, 65621, 65693, 65869, 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This is the way I worked :
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In the sequel, a squarefree odd semiprime will be called a SOSP.

 
The set of all  SOSPs will be denoted by  Z.
Let's start with taking a finite subset  A  of  Z  such that the sum  t(A)  of all reciprocals of  A is greater than  1.
Then take a subset  B  of  A  and  Let  t(B)  be the sum of the reciprocals of the members of  B.
If  t(B) > 1, select from  B  an element  b  such that  t(B) - 1/b  has a denominator as small as possible.
Replace  B  by  B \ {b}.  We repeat that process till the sum of the reciprocals is no longer bigger than  1.  If  t(B) = 1  you are lucky !
If  t(B) < 1  we select from  A \ B  an element  a  that is not in  B  such that  t(B) + 1/a   has a denominator as small as possible.
Replace  B  by  joining  a  to  B  We repeat this until the sum of reciprocals is no longer smaller than 1.   If  t(B) = 1  you are lucky !
But in most cases this process enters into an infinite loop.  This is the case when  a = b.
Therefore, the success will depend on the choice of a good  A  and  B.
After a lot of attempts I took for  A  the set of SOSPs with greatest prime divisor  p(80) = 409  and for  B  I took the set of  SOSPs with greatest prime divisor  p(79) = 401.

So, to find a solution that way, you need more luck than intelligence ... !

Also, minimality is not ensured.
When I apply this to find Egyptian fractions with semiprime denominator that sum to  1  I find a set of  111  members.
In this site solutions of  47  members are presented by Watanabe.


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