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A360179 a(1) = 1. Thereafter if a(n-1) is a novel term, a(n) = d(a(n-1)); otherwise a(n) = a(n-1) + d(u), where d is the divisor function A000005 and u is the smallest unstarred prior term (each time we use a prior term we star it, and starred terms cannot be reused).

%I A360179 #48 Apr 16 2023 09:51:16
%S A360179 1,1,2,2,3,2,4,3,5,2,4,6,4,6,8,4,7,2,5,7,10,4,7,10,12,6,8,12,16,5,9,3,
%T A360179 5,7,9,11,2,4,6,9,13,2,4,6,9,13,15,4,8,11,15,19,2,5,7,9,11,14,4,7,10,
%U A360179 12,15,18,6,10,14,18,22,4,8,11
%N A360179 a(1) = 1. Thereafter if a(n-1) is a novel term, a(n) = d(a(n-1)); otherwise a(n) = a(n-1) + d(u), where d is the divisor function A000005 and u is the smallest unstarred prior term (each time we use a prior term we star it, and starred terms cannot be reused).
%C A360179 Whilst the definition is subtly different from that of A345147, d(u) being used in place of u, the scatterplots are remarkably different, the one for this sequence displaying numerous precipitous "gorges" which are open to explanation. 1 is the only number which occurs precisely twice, all other numbers are repeated infinitely many times.
%C A360179 From _Michael De Vlieger_, Apr 04 2023: (Start)
%C A360179 The sequence is a series of nondecreasing cycles that reach a maximum M and then reset to start a new cycle. (See scatterplot B.)
%C A360179 The sequence is dynamic and responds to a bank of copies of the same number called a "prevailing low" L. When M < L, the sequence experiences a run of short or "crashed" cycles that make no headway at eliminating the copies of L, resulting in a "gorge" in the scatterplot.
%C A360179 Referring to scatterplot A:
%C A360179 The green line represents the smallest missing number u and is not actually a feature of the sequence. The red line represents the "prevailing low" L(n), also is not a feature of the sequence.
%C A360179 Dark blue terms a(n) = tau(a(n-1))..421 populate a "semi-coherent" phase (1A) of cycle c(i), where tau(n) = A000005(n).
%C A360179 Light blue terms a(n) = 422..L populate the "coherent" phase (1B) of cycle c(i). Black terms m > L populate phase (2) of c(i).
%C A360179 Magenta terms constitute a crashed cycle that has M < L; multiple consecutive crashed cycles constitute a gorge. In crashed cycles, we have only phase (1).
%C A360179 The "triple point" of the graph, where we first have phase (1B), appears to be a(14478) = 414, but is in actuality (given 2^20 terms) a(14786) = 422. (End)
%H A360179 Michael De Vlieger, <a href="/A360179/b360179.txt">Table of n, a(n) for n = 1..40000</a> [The exceptionally large b-file was added at my request. - _N. J. A. Sloane_, Apr 07 2023]
%H A360179 Michael De Vlieger, <a href="/A360179/a360179.png">Scatterplot A of a(n)</a>, n = 1..40000.
%H A360179 Michael De Vlieger, <a href="/A360179/a360179_1.png">Scatterplot B of a(n)</a> for n = 1..128. Records appear in red, local minima in blue, terms instigated by a(n) = m new to the sequence appear in gold, otherwise in green. The magenta line indicates the smallest missing number u not in a(1..n-1).
%H A360179 Michael De Vlieger, <a href="/A360179/a360179_2.png">Scatterplot of a(n)</a>, n = 1500000.
%e A360179 a(2) = 1 since a(1) = 1 is a novel term and d(1) = 1. Thus the sequence starts 1,1 and since a(2) is a repeated term, a(3) = a(2) + d(1) (1 = least unstarred prior term). Therefore a(3) = 1 + 1 = 2.
%t A360179 nn = 120; c[_] := False; h[_] := 0; f[n_] := DivisorSigma[0, n]; a[1] = j = u = 1; Do[If[c[j], k = j + f[u]; h[j]++; h[u]--, k = f[j]; c[j] = True; h[j]++]; u = Min[u, j]; Set[{a[n], j}, {k, k}]; While[h[u] == 0, u++], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, Feb 02 2023 *)
%Y A360179 Cf. A000005, A343887, A345147.
%Y A360179 Cf. A362127 records, A362128 indices of records.
%Y A360179 Cf. A362129 a(n) mod 2, A362130 d(a(n)) mod 2.
%Y A360179 Cf. A362131 smallest missing number in a(1..n).
%Y A360179 Cf. A362134 novel terms, A362135 indices of novel terms.
%Y A360179 Cf. A362136 row lengths, if this sequence seen as rows of strictly increasing terms.
%Y A360179 See A361511 for another version.
%K A360179 nonn,look,nice
%O A360179 1,3
%A A360179 _David James Sycamore_, Jan 29 2023