Notes on A372865. Michael Thomas De Vlieger, St. Louis, Missouri 202410271700. Let a(1) = 1. For n > 1, [A] a(n–1) ∈ A055932 implies a(n) = least p ≠ a(i), i < n, prime p; [B] a(n–1) ∉ A055932 implies a(n) = least k = m × q ≠ a(i), i < n, where q = A079068(a(n)) = prime q < gpf(a(n)) such that q ∤ a(n–1). 1.) Condition [A]: a(n–1) with primorial kernel implies a(n) is the smallest prime not already a term. 2.) Condition [B]: a(n–1) with squarefree kernel not a primorial implies a(n) is the smallest novel multiple m × q of the greatest prime q < gpf(a(n–1)) such that q does not divide a(n–1). 3.) The sequence presents a seemingly predictable behavior for n < 4318 (see Appendix Table A). This transforms in subtle steps for n = 4319..99527, more conspicuously in scatterplot for n = 99528..155219, intensifying for n > 222811 until it seems we never see condition [A] for n > 2048704. Phases are described below. For this reason we refer to A372865 and one like it (A372368) as "tipping point sequences", though the transformation unfolds over many iterations. 4.) Phase I. For n < 4318, the sequence features cycles c that begin with condition [A] output, i.e., prime a(n) = p. Thereafter, steps in the cycle feature strictly decreasing q as n increases through condition [B] until a(n–1) is in A059932. Then [A] commences the next cycle with the smallest missing prime p. In the first cycles, m increases as q decreases, since by definition, the prime p that starts the cycle has m = 1. Example of a cycle: c(9) through the first term of c(10). n a(n) 2 3 5 7 11 13 17 19 23 29 m q ----------------------------------------------- 33 23 . . . . . . . . 1 (1) 23 34 38 1 . . . . . . 1 2 19 35 51 . 1 . . . . 1 3 17 36 52 2 . . . . 1 4 13 37 55 . . 1 . 1 5 11 38 42 1 1 . 1 6 7 39 30 1 1 1 6 5 40 29 . . . . . . . . . 1 (1) 29 We might be led to believe that cycle c(i) begins with prime(i). For > 4316, this is not the case. 5.) This sequence exhibits tipping point behavior induced by onset of gpf(m) > q, which resets the cycle, delays closure, and increments m(q) of repeated primes q. 6.) Instances of primes resulting from [B] first occurs at c(4318) = prime(71). These condition [B] primes rejuvenate a cycle that might otherwise close. We call these rejuvenated cycles "extended cycles". n a(n) 2 3 5 7 11 13 17 19 ~ 347 349 353 359 m pi(q) ----------------------------------------------------------------- 4313 2793 . 1 . 2 . . . 1 147 8 4314 2754 1 4 . . . . 1 162 7 4315 2457 . 3 . 1 . 1 189 6 4316 2464 5 . . 1 1 224 5 4317 1795 . . 1 . . . . . ~ . . . 1 [359] 3 4318 353 . . . . . . . . ~ . . 1 1 71 4319 1047 . 1 . . . . . . ~ . 1 3 70 4320 2776 3 . . . . . . . ~ 1 8 69 7.) Phase II A. For n > 4316, extended cycles with repeated q become increasingly common, therefore certain m(q) become large compared to other m(q) such that gpf(m(q)) > gpf(a(n–1)) and eventually, exceed q for relatively large q such that condition [A] becomes rare. 8.) Phase II B. Multiplier m(q) rivals primes in the vicinity of q = prime(146) = 839 in size. This induces repetitively rejuvenating cycles beginning at n = 144128. A conspicuous swelling feature in scatterplot appears around n = 144128..222811. For n = 99528..155219, we have only composite numbers. a(99527) = 1097 = prime(184), a(155820) = 1103 = prime(185). For n = 155820..222811, there are 78 primes, 10 of which through condition [A]. 9.) Phase II C. Starting with a(262796) = prime(265) = 1697, we develop a ridge of high multiples, for example, m(q) = 1448 for q = prime(226) = 1429, with a steep gradient of declining multiples m(q) beginning with the following: m(q) = 1367 for q = prime(227) = 1433. For n = 222811..262605, we have only composite numbers. a(222811) = 1669 = prime(263), a(262605) = 1693 = prime(264). See Appendix Tables B and C comparing gpf(a(n–1)) with m(q) in Phase I and Phase II. 9.) Phase III. The last cycle observed is c(135) beginning at a(2048704) = prime(742) = 5647, which has not closed after 2^25 = 33554432 terms. For n > 2048704, repetitively rejuvenating cycles have established multiples m(q) that are at least as large as q for q = prime(662) = 4951. Primes continue to enter the sequence through condition [B] throughout, but it appears the cycle cannot close. As primes continue to enter and as multiplicity builds for primes only a few dozen primes smaller than the record primes, it becomes ever increasingly likely that the cycle will be rejuvenated rather than proceed to find a number in A055932 (effectively, a number with greatest prime factor 11) and close the cycle. 10.) Since there are 135 full cycles in the sequence, the intersection of a(1..2^25) and A055932 has 136 terms, the largest of which is A055932(157) = 11550, a number with 5 distinct prime factors. Those terms in A055932 less than A055932(157) missing from A376865 are listed below. {6144, 6912, 7680, 7776, 8100, 8192, 8640, 8748, 9000, 9216, 9450, 9600, 9720, 10080, 10290, 10368, 10500, 10800, 11250, 11340, 11520}. Appendix: Table A: Prime decomposition of the first 100 terms, with (*) in m representing Condition [A]: p 11112233444 n a(n) 235713793917137 m π(q) ----------------------------------------- 1 1 . * 0 2 2 1 * 1 3 3 .1 * 2 4 4 2 2 1 5 5 ..1 * 3 6 6 11 2 2 7 7 ...1 * 4 8 10 1.1 2 3 9 9 .2 3 2 10 8 3 4 1 11 11 ....1 * 5 12 14 1..1 2 4 13 15 .11 3 3 14 12 21 4 2 15 13 .....1 * 6 16 22 1...1 2 5 17 21 .1.1 3 4 18 20 2.1 4 3 19 18 12 6 2 20 17 ......1 * 7 21 26 1....1 2 6 22 33 .1..1 3 5 23 28 2..1 4 4 24 25 ..2 5 3 25 24 31 8 2 26 19 .......1 * 8 27 34 1.....1 2 7 28 39 .1...1 3 6 29 44 2...1 4 5 30 35 ..11 5 4 31 27 .3 9 2 32 16 4 8 1 33 23 ........1 * 9 34 38 1......1 2 8 35 51 .1....1 3 7 36 52 2....1 4 6 37 55 ..1.1 5 5 38 42 11.1 6 4 39 30 111 6 3 40 29 .........1 * 10 41 46 1.......1 2 9 42 57 .1.....1 3 8 43 68 2.....1 4 7 44 65 ..1..1 5 6 45 66 11..1 6 5 46 49 ...2 7 4 47 40 3.1 8 3 48 36 22 12 2 49 31 ..........1 * 11 50 58 1........1 2 10 51 69 .1......1 3 9 52 76 2......1 4 8 53 85 ..1...1 5 7 54 78 11...1 6 6 55 77 ...11 7 5 56 45 .21 9 3 57 32 5 16 1 58 37 ...........1 * 12 59 62 1.........1 2 11 60 87 .1.......1 3 10 61 92 2.......1 4 9 62 95 ..1....1 5 8 63 102 11....1 6 7 64 91 ...1.1 7 6 65 88 3...1 8 5 66 56 3..1 8 4 67 50 1.2 10 3 68 48 41 16 2 69 41 ............1 * 13 70 74 1..........1 2 12 71 93 .1........1 3 11 72 116 2........1 4 10 73 115 ..1.....1 5 9 74 114 11.....1 6 8 75 119 ...1..1 7 7 76 104 3....1 8 6 77 99 .2..1 9 5 78 63 .2.1 9 4 79 60 211 12 3 80 43 .............1 * 14 81 82 1...........1 2 13 82 111 .1.........1 3 12 83 124 2.........1 4 11 84 145 ..1......1 5 10 85 138 11......1 6 9 86 133 ...1...1 7 8 87 136 3.....1 8 7 88 117 .2...1 9 6 89 110 1.1.1 10 5 90 70 1.11 10 4 91 54 13 18 2 92 47 ..............1 * 15 93 86 1............1 2 14 94 123 .1..........1 3 13 95 148 2..........1 4 12 96 155 ..1.......1 5 11 97 174 11.......1 6 10 98 161 ...1....1 7 9 99 152 3......1 8 8 100 153 .2....1 9 7 ----------------------------------------- n a(n) 111122334445 m π(q) 2357137939171373 Table B. Graphs comparing prime g = gpf(a(n–1)) with multiplier m(q), representing the former by "o" and the latter by ".". Phase I behavior: m is small compared to g, more significantly, gpf(m) < g. n = 37..61: 10 20 30 40 50 g m(q) . | . | . | . | . | ------------------------------------------------------------ 11 7 . o 5 9 o . 2 16 o . 37 1 o 31 2 . o 29 3 . o 23 4 . o 19 5 . o 17 6 . o 13 7 . o 11 8 . o 7 8 o. 5 10 o . 3 16 o . 41 1 o 37 2 . o 31 3 . o 29 4 . o 23 5 . o 19 6 . o 17 7 . o 13 8 . o 11 9 . o 7 9 o . 5 12 o . Table C: Phase II behavior: m and g are comparable; at times when m exceeds g, m is also prime and rejuvenates the cycle. Passing over the same g again and again increment m each time, building a "ridge" of primes q with high m(q). Shown below is a particularly compressed succession of rejuvenations that occur near the visible portion in log log scatterplot where there seems to be a large gap in small values around n = 155820. It is the very first in a series of repeated episodes of compressed rejuvenations. n = 144135..144159: 820 830 840 850 860 g m(q) . | . | . | . | . | ------------------------------------------------------------ 857 812 . o 853 817 . o 839 830 . o 829 840 o . 827 847 o . 823 851 o . 853 821 . o 839 831 . o 829 841 o . 827 848 o . 823 852 o . 821 854 o . 859 811 o 857 813 . o 853 818 . o 839 832 . o 829 842 o . 827 849 o . 853 823 . o 839 833 . o 829 843 o . 827 850 o . 823 854 o . 821 855 o . 811 860 .