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A364236 a(1) = 1. For n > 1, if a(n-1) is a novel term, a(n) = d(a(n-1)), else if a(n-1) is a repeat term seen k (>1) times, a(n) = a(n-1) + d(k-1), where d is the divisor counting function A000005.

%I A364236 #22 Jul 15 2023 08:49:11
%S A364236 1,1,2,2,3,2,4,3,4,5,2,4,6,4,6,7,2,5,6,8,4,7,8,9,3,5,7,9,10,4,6,8,10,
%T A364236 11,2,4,8,10,12,6,9,11,12,13,2,6,8,11,13,14,4,6,10,12,14,15,4,8,10,13,
%U A364236 15,16,5,7,9,11,13,15,17,2,4,7,10,12,14,16,17,18
%N A364236 a(1) = 1. For n > 1, if a(n-1) is a novel term, a(n) = d(a(n-1)), else if a(n-1) is a repeat term seen k (>1) times, a(n) = a(n-1) + d(k-1), where d is the divisor counting function A000005.
%C A364236 In other words the appearance of a novel term m introduces d(m) as next term, whereas the appearance of repeat term m introduces m + the number of divisors of the number of repetitions of m.
%C A364236 1 is seen only twice, but all other numbers appear infinitely many times.
%C A364236 Prime terms may appear in 3 different ways: consequent to the second appearance of p-1, to the first appearance of m where d(m) = p, or to a repeat term m seen k (>1) times, where m + d(k-1) = p.
%C A364236 The plot consists of consecutive strictly increasing trajectories starting d(r(k)) after the k_th record term r(k), and ending with r(k+1) = r(k)+1, meaning that records are given by A000027. This behavior, which determines the smooth whaleback shape of the plot is open to explanation.
%H A364236 Michael De Vlieger, <a href="/A364236/b364236.txt">Table of n, a(n) for n = 1..10000</a>
%H A364236 Michael De Vlieger, <a href="/A364236/a364236.png">Plot of a(n)</a>, n = 1..256, showing chains c(k) beginning with tau(k-1) and strictly increasing until we reach k itself. We highlight chain minimum tau(k-1) in blue and maximum k in red.
%H A364236 Michael De Vlieger, <a href="/A364236/a364236_1.png">Plot of a(n)</a>, n = 1..2^16, showing fine structure.
%H A364236 Michael De Vlieger, <a href="/A364236/a364236_2.png">Plot of a(n)</a> n = 1..2^20, showing aggregate structure.
%e A364236 a(1) = 1 is a novel term so a(2) = d(a(1)) = d(1) = 1.
%e A364236 Since 1 has been repeated once, a(3) = 1 + d(1) = 2.
%e A364236 a(3) = 2 introduces a(4) = d(2) = 2 and so on.
%t A364236 nn = 120; c[_] := 0; a[1] = j = 1; f[x_] := DivisorSigma[0, x]; Do[k = If[# == 0, f[j], j + f[#] ] &[c[j]]; c[j]++; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, Jul 14 2023 *)
%o A364236 (PARI) lista(nn) = {my(va=vector(nn)); va[1] = 1; for (n=2, nn, my(vv = Vec(va, n-1)); my(k = #select(x->(x==va[n-1]), vv)); if (k==1, va[n] = numdiv(va[n-1]), va[n] = va[n-1] + numdiv(k-1));); va;} \\ _Michel Marcus_, Jul 14 2023
%Y A364236 Cf. A000005, A000027, A345147, A360179.
%K A364236 nonn,look
%O A364236 1,3
%A A364236 _David James Sycamore_, Jul 14 2023
%E A364236 More terms from _David A. Corneth_, Jul 14 2023