Linear Relationships Between Loeschian Numbers

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 $x^2+xy+y^2$ with $0\le x\le y$.

We see that: $$\begin{array}{rl}{7}^2+7\times 161+{161}^2& =27097\\ {47}^2+47\times 136+{136}^2& =27097\\ {64}^2+64\times 123+{123}^2& =27097\\ {77}^2+77\times 112+{112}^2& =27097\end{array}$$The first member of the sequence is the famous taxi cab number 1729 and I've written about this in my post titled Loeschian Numbers on January 5th 2022.

I then decided to investigate what numbers have exactly five representations by the quadratic form $x^2+xy+y^2$ with $0\le x\le y$. The initial numbers that satisfy this condition are 8281, 17689, 24843, 31213, 33124, 45619, 47089, 53067, 56203, 56791, 57967, 62377, 63973, ...

For example 8281 has $(x,y)$ representations as:

(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 $(x,y)$ representations as:

(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 $(x,y)$ representations as:

(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 $(x,y)$ representations as:

(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: $10374x+43225$

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:$$\begin{gathered} 10374=2\times 3\times 7\times 13\times 19 \\ 43225=5^2\times 7\times 13\times 19 \end{gathered}$$Furthermore, we have:$$\begin{array}{rl}10374\times 1+43225& =53599\\ 10374\times 2+43225& =63973\\ 10374\times 3+43225& =74347\\ 10374\times 4+43225& =84721\end{array}$$However, using my Jupyter notebook, I extended the range of terms and found that the very next term after 84721 did not match up. The next term is 104377 but$$10374\times 5+43225=95095\ne 104377$$Later terms however, do match up even though, term by term, the two series diverge sharply (see Figure 1) .

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

54 out the 346 terms that I generated match up with the linear sequence. That's almost 20% of the terms. There's clearly more at play here than meets the eye but I don't know what it is. If we look at the list of matching terms, we find the following values for $x$ in $10374x+43225$ generate the terms in the sequence:

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 17, 19, 21, 22, 23, 24, 26, 28, 29, 31, 33, 34, 36, 41, 42, 43, 47, 48, 51, 52, 54, 56, 57, 58, 59, 62, 64, 66, 68, 69, 72, 73, 77, 79, 83, 86, 87, 91, 92, 97, 113, 116, 123

Thus for the last term we have $10374\times 123+43225=1319227$.