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A345147 a(1)=1. If a(n) is a novel term, a(n+1) = A000005(a(n)). If a(n) is a repeat term, a(n+1) = a(n)+m, where m is the least prior term (which once used in this way is not used again).

%I A345147 #21 Jan 12 2024 10:05:33
%S A345147 1,1,2,2,3,2,4,3,5,2,4,6,4,7,2,5,7,11,2,6,8,4,8,12,6,11,16,5,11,16,22,
%T A345147 4,10,4,8,12,19,2,9,3,5,8,13,2,10,12,20,6,14,4,10,14,22,31,2,12,14,24,
%U A345147 8,18,6,14,20,31,42,8,19,27,4,16,20,32,6,18,24,36
%N A345147 a(1)=1. If a(n) is a novel term, a(n+1) = A000005(a(n)). If a(n) is a repeat term, a(n+1) = a(n)+m, where m is the least prior term (which once used in this way is not used again).
%C A345147 Let conditions 0 and 1 respectively pertain to novel a(n) and extant a(n) respectively. For n>3 condition 0 results in a fall from a(n) to a(n+1), whereas condition 1 results in an increase, which may be sustained over several consecutive terms as the least unused term is applied to the result of a sum which is the same as a prior term, therefore prompting a further application of condition 1, until a novel term is reached.
%C A345147 A distinguishing feature of the scatterplot of this sequence is that the data are highly concentrated around several quasi-radial, apparently straight line "spokes" emanating from the origin, which appear to be equally spaced. The lower spoke relates to applications of condition 0, producing the values d(k) of novel terms k, which being small by comparison to k, are distributed widely across the lower zone of the plot. Higher spokes relate to consecutive terms generated by condition 1, namely repeated addition of the least unused term m.
%C A345147 Since m appears in multiples and increments by small amounts, successive additions result in roughly equally spaced consecutive terms. Repeated application of condition 1 continues until a novel term k is reached, followed by a fall to d(k), from which the process repeats, carrying forward any remaining copies of m until all are used, at which point m increments, causing the spokes to diverge gradually as the sequence advances.
%C A345147 Records beyond a(1) arise consequent only to repetitions of condition 1, and not to the appearance of a novel term. With expansion of the data the spokes are gradually obscured from the bottom up by horizontal striations resulting from the fact that many different numbers have the same number of divisors, and that with greater values of a(n), greater values of d(a(n)) can appear. As the lower spokes are obscured, so new spokes appear above, corresponding to new records.
%C A345147 The greatest fall from a(n) to a(n+1) is when a(n) is a prime record, then a(n+1)=2. This sometimes creates a remarkable disturbance in the sequence, involving many chaotic reversals before recovery to the more stable pattern described above is achieved. The first conspicuous example of this is at a(243)=89. A repeat of condition 0 can only be consequent to a novel highly divisible term k for which d(k) is also novel. For n<=2^19 the only occurrence of this is at a(422)=144, followed by d(144)=15. For more details see links.
%H A345147 Michael De Vlieger, <a href="/A345147/b345147.txt">Table of n, a(n) for n = 1..10001</a>
%H A345147 Michael De Vlieger, <a href="http://vincico.com/seq/a345147.html">The Cliffside Sequence</a>, analysis of this sequence with code, 2021.
%H A345147 Michael De Vlieger, <a href="/A345147/a345147.png">Scatterplot of a(n)</a> for n = 1..2^18 showing quasi-radial striations and "chatter" in catastrophic episodes.
%H A345147 Michael De Vlieger, <a href="/A345147/a345147_1.png">Annotated scatterplot of a(n)</a> for n = 1..2^10 showing records in red, local minima in blue, m resulting from the novel condition, thus d(a(n)) in gold, and m resulting from the extant condition in green.
%H A345147 Michael De Vlieger, <a href="/A345147/a345147_2.png">Scatterplot of a(n)</a> for n = 1..2^12 labeling the first indices of extant-condition catastrophes.
%e A345147 a(1)=1 is a novel term so a(2)=d(1)=1. Since a(2) is a repeat term and 1 is the least unused prior term, a(3)=a(2)+1=2. Then since 2 is a novel term, a(4)=d(a(3))=d(2)=2; and so on.
%t A345147 Block[{a = {1}, s = {}}, Do[If[FreeQ[#2, #1], AppendTo[a, DivisorSigma[0, a[[-1]]] ], AppendTo[a, a[[-1]] + First[s] ]; Set[s, Rest@ s]] & @@ {First[#1], #2} & @@ TakeDrop[a, -1]; Set[s, Insert[s, a[[-2]], LengthWhile[s, # < a[[-2]] &] + 1]], 105]; a] (* _Michael De Vlieger_, Jun 15 2021 *)
%Y A345147 Cf. A000005, A002182, A002183, A005179, A343887.
%Y A345147 Cf. A360179.
%K A345147 nonn
%O A345147 1,3
%A A345147 _David James Sycamore_ and _Michael De Vlieger_ Jun 09 2021