The 4k+1 versus 4k+3 Composite Race

It was only recently that I posted about the \(4k-1\) versus \(4k+1\) Prime Race in connection with my diurnal age of 26833 and today, aged 26835 days, I find myself involved in another race, this time the \(4k+1\) versus \(4k+3\) Composite Race. Of course \(4k+3\) is the same as \(4k-1\) but I'll stick with the former as that's the way the OEIS entry is described:

A093180

Odd composites (including 1 in the count) where the number 1 mod 4 equals the number 3 mod 4.

The sequence runs:

26835, 26851, 26855, 26859, 26867, 26871, 26875, 26883, 26887, 26895, 26899, 26923, 616771, 616775, 616779, 616795, 616831, 616835, 616839, 616847, 616853, 616857, 616861, 616865, 616869, 616875, 616881, 616885, 616889, 616893, 617013, 617017, 617021, 617025, 617031, 617035, 617259, 617263, 617267, 617329, 617361, 617367, 617373, 617377, 617381, 617385, 617391, 617395, 617399, 617405, 617409, 617415, 617419, 617423, 617427, 617433, 617437, 617441, 617445, 617451, 617457, 617461, 617465, 617475, 617511, 617515, 617519, 617525, 617529, 617535, 617541, 617545, 617549, 617553, 617557, 617561, 617565, 617569, 617573, 617577, 617583, 617683, 617687, 617701, 617705, 617711, 617715, 618523, 618527, 618531, 618535, 618539, 618543, 618595, 618599, 618603, 618607, 618611, 618615, 618639, 622795, 622799, 622803, 622807, 622811, 622817, 622821, 622825, 622829, 622833, 622837, 622841, 622845, 622989, 622993, 623005, 623109, 623113, 623117, 623121, 623125, 623129, 623133, 623137, 623141, 623145, 623149, 623153, 623157, 623161, 623165, 623169, 623175, 623179, 623183, 623187, 623191, 623195, 623199, 623203, 623207, 623213, 623217, 623301, 623307, 623311, 623315, 623319, 623325, 623331, 623335, 623339, 623345, 623349, 623357, 623361, 623365, 623369, 623373, 623377, 623381, 623395, 623399, 623405, 623409, 623413, 623425, 623429, 623435, 623441, 623445, 623449, 623453, 623457, 623461, 623465, 623469, 623473, 623593, 623597, 623601, 623605, 623609, 623613, 623805, 623809, 623813, 623817, 623821, 623825, 623829, 623833, 623837, 623843, 623847, 623895, 623899, 623903, 623907, 623911, 623915, 623919, 623931, 623937, 623941, 623945, 623951, 623955, 623961, 623967, 623971, 623975, 623981, 623987, 623993, 623997, 624001, 624005, 624011, 624015, 624019, 624023, 624027, 624039, 624043, 624051, 624055, 624059, 624063, 624091, 624095, 624101, 624105, 624109, 624113, 624117, 624123, 624127, 624131, 624137, 624143, 624147, 624153, 624157, 624161, 624167, 624171, 624175, 624179, 624183, 624187, 624235, 624239, 624245, 624249, 624255, 626923, 626927, 626933, 626937, 626941, 626945, 626951, 626957, 627043, 627047, 627051, 627055, 627351, 627357, 627361, 627365, 627369, 627373, 627381, 627387, 627435, 627439, 627443, 627447, 627453, 627457, 627461, 627465, 627469, 627473, 627477, 627485, 627489, 627495, 627499, 627503, 627507, 627543, 627723, 627727, 627731, 627737, 627741, 627745, 627793, 627801, 627805, 627809, 627815, 627819, 627823, 627827, 627831, 627835, 627839, 627845, 627849, 627853, 627857, 627863, 627867, 627871, 627875, 627879, 627883, 627887, 627891, 627895, 627899, 627905, 627909, 627915, 627975, 627979, 627983, 627987, 627991, 627995, 627999, 628003, 628007, 628011, 628017, 628185, 628401, 628405, 628409, 628413, 628417, 628421, 628923, 628927, 628931, 628935, 628943, 628947, 628951, 628955, 628959, 628963, 628967, 628971, 628977, 628981, 628985, 628989, 629025, 629061, 629065, 629069, 629073, 629077, 629245, 629265, 629269, 629273, 629277, 629313, 629317, 629321, 629325, 629329, 629333, 629337, 629345, 629349, 629355, 629359, 629363, 629367, 629403, 629407, 629419, 629423, 629427, 629433, 629437, 629441, 629445, 629469, 629473, 629477, 629481, 629487, 629511, 629517, 629521, 629525, 629529, 629533, 629725, 629729, 629733, 629745, 629751, 629755, 629759, 629763, 629775, 630715, 630735, 630741, 630745, 630749, 630753, 630757, 630761, 630765, 630769, 630773, 630777, 630781, 630785, 630789, 630793, 630805, 630809, 630813, 630817, 630821, 630843, 630847, 630851, 630855, 630859, 632223, 632563, 632567, 632571, 632575, 632579, 632583, 632587, 632611, 632615, 632619, 632679, 632719, 632723, 632727, 632731, 632735, 632739, 632779, 632783, 632787, 632791, 632795, 632799, 632803, 632807, 632811, 632817, 632821, 632825, 632829, 632833, 632837, 632883, 632887, 632891, 632895, 632901, 632905, 632909, 632915, 632919, 633003, 633007, 633011, 633017, 633021, 633025, 633029, 633033, 633081, 633085, 633089, 633095, 633099, 633103, 633107, 633111, 633115, 633119, 633123, 633127, 633131, 633137, 633141, 633145, 633149, 633155, 633159, 633165, 633169, 633173, 633177, 633181, 633185, 633191, 633195, 633201, 633205, 633309, 633313, 633385, 633389, 633393, 633397, 633409, 633413, 633417, 633421, 633425, 633431, 633435, 633439, 633443, 633447, 633453, 633457, 633465, 633489, 633493, 633561, 633565, 633573, 633577, 633581, 633587, 633591, 633595, 633615, 633619, 633631, 633635, 633639, 633643, 633647, 633669, 633673, 633677, 633681, 633685, 633689, 633693, 633697, 633701, 633705, 633709, 633713, 633717, 633721, 633725, 633729, 633733, 633737, 633743, 633747, 633759, 633763, 633783, 633787, 633795, 633879

Here is a permalink to the algorithm that I used to generate these numbers. Interestingly 26835 is the very first member of the sequence while 633879 is the last in the range up to one million. The graph of the count looks pretty much the same as the one in the Prime Race post. See Figure 1 where is the plot is from 1 to one million with one million marked as 1.0 and 0.6 representing 600,000 etc.

Figure 1: permalink