It all began innocently enough with my investigation of the number associated with my diurnal age today, 27097. It turns out that this number is a member of OEIS A198775:
A198775 | Numbers having exactly four representations by the quadratic form |
We see that:
I then decided to investigate what numbers have exactly five representations by the quadratic form
For example 8281 has
(0, 91), (11, 85), (19, 80), (39, 65), (49, 56)
The sequence is not recognised by the OEIS. I then moved to numbers that have six such representations. This yielded 12103, 19747, 22477, 23569, 27391, 28861, 32851, 34447, 36309, 36673, 38857, 40033, 42679, 43771, 46501, 48412, 50323, 50869, ...
For example, 12103 has
(2, 109), (21, 98), (27, 94), (34, 89), (49, 77), (61, 66)
Again, this sequence is not recognised by the OEIS. I then turned to numbers that have seven such representations. This yielded 104377 and 105469 although I'm sure there are more if I extended the search.
For example, 104377 has
(47, 297), (69, 283), (72, 281), (107, 256), (131, 237), 137, 232), (181, 192)
Again this sequence is not recognised by the OEIS. I then turned to numbers that have eight such representation. This yielded 53599, 63973, 74347, 84721, ...
For example, 53599 has
(3, 230) (25, 218) (43, 207) (58, 197) (85, 177) (90, 173) (102, 163) (122, 145)
Again this sequence wasn't recognised by the OEIS but I got the following intriguing message:
Sorry, but the terms do not match anything in the table.
Your sequence appears to be:
This rather shocked me but it's indeed true. I checked the factorisation of the two numbers and as can be seen they are highly factorisable:
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Figure 1 |
Actual Series:
53599, 63973, 74347, 84721, 104377, 105469, 115843, 121303, 126217, 136591, 138229, 144781, 152551, 160797, 164983, 167713, 172081, 177289, 178087, 188461, 189007, 191737, 191919, 202027, 205387, 205933, 211603, 214396, 219583, 222859, 223041, 225589, 238693, 240331, 241129, 245791, 251503, 254163, 255892, 261079, 262171, 265993, 271453, 271999, 273637, 276241, 280231, 281827, 283309, 285649, 290563, 292201, 297388, 298753, 300181, 300979, 307489, 309127, 312949, 313131, 316407, 325507, 325717, 326599, 329251, 329707, 333697, 338884, 344071, 345247, 346801, 347529, 348859, 352261, 358267, 359233, 363909, 364819, 367003, 371917, 378651, 379561, 383173, 385567, 388759, 392119, 392977, 393421, 395941, 397537, 399931, 403039, 405223, 405769, 409773, 414687, 416689, 417487, 417508, 421876, 424669, 425971, 426517, 427609, 432523, 434343, 436639, 438529, 442897, 445081, 449407, 451801, 457219, 457653, 459277, 459823, 463372, 467077, 468013, 468559, 475741, 476749, 477337, 478933, 481663, 482391, 485212, 486031, 489307, 489769, 494949, 496951, 503139, 504868, 506863, 508417, 510601, 513019, 514843, 516243, 523621, 524797, 528619, 530803, 531867, 534261, 536389, 537943, 538447, 541177, 543571, 543907, 544453, 545713, 546364, 548821, 552916, 554743, 557479, 558961, 561379, 565383, 566293, 567021, 569023, 569191, 571753, 572299, 573097, 575211, 575757, 579124, 582673, 586117, 589057, 590863, 591409, 592249, 595231, 603421, 604903, 605059, 606081, 609427, 610204, 612313, 614341, 616161, 617799, 619801, 622573, 623371, 624169, 624967, 625177, 627991, 629083, 632149, 634291, 634543, 634809, 637819, 640927, 641173, 643188, 644371, 644917, 646009, 647311, 647881, 648613, 649831, 650503, 655291, 658749, 659932, 661297, 664573, 667147, 668577, 669123, 670033, 670852, 671251, 672049, 673309, 676767, 679357, 679861, 681163, 684019, 684229, 686413, 688324, 697333, 700063, 703969, 704977, 707161, 708253, 709156, 710353, 710437, 711607, 712348, 712411, 716079, 720993, 721981, 723387, 724087, 726313, 726817, 727909, 737149, 737373, 740467, 742729, 743071, 746179, 746263, 748657, 750841, 753571, 753844, 754509, 755209, 756028, 759031, 762489, 764491, 766948, 767011, 767676, 771043, 771589, 772597, 774151, 774319, 775333, 776209, 778687, 778813, 781417, 783237, 783601, 786513, 786961, 788671, 790153, 794941, 795739, 797979, 798343, 800527, 802123, 807079, 808108, 811447, 813001, 814359, 815997, 820477, 820911, 821548, 823732, 825643, 826987, 827659, 828723, 834613, 835639, 838201, 839059, 840007, 840693, 842023, 843661, 845481, 846412, 847483, 849927, 850297, 856947, 857584, 860587, 862771, 863569, 864367, 866047, 866761, 869953, 871507, 871689, 872599, 875161, 876603, 878332, 879823, 882973, 884317, 886483, 886879, 887341, 890701, 891436, 892164, 896077, 896259, 897883, 899689, 900543, 902356, 902937, 904267, 904813, 905107, 905863, 909181, 909571, 912457, 913003, 917833, 919429, 920647, 922467, 923377, 926443, 927381, 932197, 932659, 933751, 935389, 937099, 938119, 938847, 939393, 940849, 941317, 941773, 943033, 944167, 945763, 949221, 949753, 950677, 952861, 954772, 957229, 961093, 961324, 962143, 964516, 966301, 971299, 972439, 976521, 977151, 979279, 979797, 983164, 984067, 984529, 987259, 987753, 988183, 989121, 993643, 993811, 994357, 996151, 997633, 999037, 999271, 1000027, 1001091, 1002589, 1003093, 1004731, 1009489, 1016652, 1019179, 1019683, 1021657, 1023568, 1030351, 1030393, 1031401, 1032213, 1035139, 1035741, 1038331, 1042587, 1042951, 1044316, 1049503, 1051687, 1052233, 1057069, 1060423, 1061347, 1062271, 1063972, 1068249, 1079029, 1093963, 1121029, 1128673, 1151059, 1177813, 1215487, 1241023, 1246609, 1319227, 1323049, 1356901
Linear Series:
53599, 63973, 74347, 84721, 95095, 105469, 115843, 126217, 136591, 146965, 157339, 167713, 178087, 188461, 198835, 209209, 219583, 229957, 240331, 250705, 261079, 271453, 281827, 292201, 302575, 312949, 323323, 333697, 344071, 354445, 364819, 375193, 385567, 395941, 406315, 416689, 427063, 437437, 447811, 458185, 468559, 478933, 489307, 499681, 510055, 520429, 530803, 541177, 551551, 561925, 572299, 582673, 593047, 603421, 613795, 624169, 634543, 644917, 655291, 665665, 676039, 686413, 696787, 707161, 717535, 727909, 738283, 748657, 759031, 769405, 779779, 790153, 800527, 810901, 821275, 831649, 842023, 852397, 862771, 873145, 883519, 893893, 904267, 914641, 925015, 935389, 945763, 956137, 966511, 976885, 987259, 997633, 1008007, 1018381, 1028755, 1039129, 1049503, 1059877, 1070251, 1080625, 1090999, 1101373, 1111747, 1122121, 1132495, 1142869, 1153243, 1163617, 1173991, 1184365, 1194739, 1205113, 1215487, 1225861, 1236235, 1246609, 1256983, 1267357, 1277731, 1288105, 1298479, 1308853, 1319227, 1329601, 1339975, 1350349, 1360723, 1371097, 1381471, 1391845, 1402219, 1412593, 1422967, 1433341, 1443715, 1454089, 1464463, 1474837, 1485211, 1495585, 1505959, 1516333, 1526707, 1537081, 1547455, 1557829, 1568203, 1578577, 1588951, 1599325, 1609699, 1620073, 1630447, 1640821, 1651195, 1661569, 1671943, 1682317, 1692691, 1703065, 1713439, 1723813, 1734187, 1744561, 1754935, 1765309, 1775683, 1786057, 1796431, 1806805, 1817179, 1827553, 1837927, 1848301, 1858675, 1869049, 1879423, 1889797, 1900171, 1910545, 1920919, 1931293, 1941667, 1952041, 1962415, 1972789, 1983163, 1993537, 2003911, 2014285, 2024659, 2035033, 2045407, 2055781, 2066155, 2076529, 2086903, 2097277, 2107651, 2118025, 2128399, 2138773, 2149147, 2159521, 2169895, 2180269, 2190643, 2201017, 2211391, 2221765, 2232139, 2242513, 2252887, 2263261, 2273635, 2284009, 2294383, 2304757, 2315131, 2325505, 2335879, 2346253, 2356627, 2367001, 2377375, 2387749, 2398123, 2408497, 2418871, 2429245, 2439619, 2449993, 2460367, 2470741, 2481115, 2491489, 2501863, 2512237, 2522611, 2532985, 2543359, 2553733, 2564107, 2574481, 2584855, 2595229, 2605603, 2615977, 2626351, 2636725, 2647099, 2657473, 2667847, 2678221, 2688595, 2698969, 2709343, 2719717, 2730091, 2740465, 2750839, 2761213, 2771587, 2781961, 2792335, 2802709, 2813083, 2823457, 2833831, 2844205, 2854579, 2864953, 2875327, 2885701, 2896075, 2906449, 2916823, 2927197, 2937571, 2947945, 2958319, 2968693, 2979067, 2989441, 2999815, 3010189, 3020563, 3030937, 3041311, 3051685, 3062059, 3072433, 3082807, 3093181, 3103555, 3113929, 3124303, 3134677, 3145051, 3155425, 3165799, 3176173, 3186547, 3196921, 3207295, 3217669, 3228043, 3238417, 3248791, 3259165, 3269539, 3279913, 3290287, 3300661, 3311035, 3321409, 3331783, 3342157, 3352531, 3362905, 3373279, 3383653, 3394027, 3404401, 3414775, 3425149, 3435523, 3445897, 3456271, 3466645, 3477019, 3487393, 3497767, 3508141, 3518515, 3528889, 3539263, 3549637, 3560011, 3570385, 3580759, 3591133, 3601507, 3611881, 3622255, 3632629, 3643003, 3653377, 3663751, 3674125, 3684499, 3694873, 3705247, 3715621, 3725995, 3736369, 3746743, 3757117, 3767491, 3777865, 3788239, 3798613, 3808987, 3819361, 3829735, 3840109, 3850483, 3860857, 3871231, 3881605, 3891979, 3902353, 3912727, 3923101, 3933475, 3943849, 3954223, 3964597, 3974971, 3985345, 3995719, 4006093, 4016467, 4026841, 4037215, 4047589, 4057963, 4068337, 4078711, 4089085, 4099459, 4109833, 4120207, 4130581, 4140955, 4151329, 4161703, 4172077, 4182451, 4192825, 4203199, 4213573, 4223947, 4234321, 4244695, 4255069, 4265443, 4275817, 4286191, 4296565, 4306939, 4317313, 4327687, 4338061, 4348435, 4358809, 4369183, 4379557, 4389931, 4400305, 4410679, 4421053, 4431427, 4441801, 4452175, 4462549, 4472923, 4483297, 4493671, 4504045, 4514419, 4524793, 4535167, 4545541, 4555915, 4566289, 4576663, 4587037, 4597411, 4607785, 4618159, 4628533, 4638907, 4649281, 4659655, 4670029, 4680403, 4690777, 4701151, 4711525, 4721899, 4732273, 4742647, 4753021, 4763395, 4773769
Matching Terms:
53599, 63973, 74347, 84721, 105469, 115843, 126217, 136591, 167713, 178087, 188461, 219583, 240331, 261079, 271453, 281827, 292201, 312949, 333697, 344071, 364819, 385567, 395941, 416689, 468559, 478933, 489307, 530803, 541177, 572299, 582673, 603421, 624169, 634543, 644917, 655291, 686413, 707161, 727909, 748657, 759031, 790153, 800527, 842023, 862771, 904267, 935389, 945763, 987259, 997633, 1049503, 1215487, 1246609, 1319227
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