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A354606 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that have the same number of divisors as a(n-1).

%I A354606 #21 Dec 12 2024 11:09:34
%S A354606 1,1,2,1,3,2,3,4,1,4,2,5,6,1,5,7,8,2,9,3,10,3,11,12,1,6,4,4,5,13,14,5,
%T A354606 15,6,7,16,1,7,17,18,2,19,20,3,21,8,9,6,10,11,22,12,4,7,23,24,1,8,13,
%U A354606 25,8,14,15,16,2,26,17,27,18,5,28,6,19,29,30,2,31,32,7,33,20,8,21,22,23,34
%N A354606 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that have the same number of divisors as a(n-1).
%C A354606 After 250000 terms the most common number of divisors of all terms are 4, 8, 2, 12, 16 divisors. These correspond to the uppermost five lines of the attached image. It is unknown if these stay the most common or are passed by numbers with more divisors as n gets arbitrarily large.
%C A354606 See A355606 for the indices where a(n) = 1.
%H A354606 Scott R. Shannon, <a href="/A354606/b354606.txt">Table of n, a(n) for n = 1..10000</a>
%H A354606 Scott R. Shannon, <a href="/A354606/a354606.png">Image of the first 250000 terms</a>. The green line is y = n.
%H A354606 Michael De Vlieger, <a href="/A354606/a354606_1.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%e A354606 a(6) = 2 as a(5) = 3 which has two divisors, and the total number of terms in the first five terms with two divisors is two, namely a(3) = 2 and a(5) = 3.
%t A354606 nn = 120; c[_] := 0; a[1] = j = 1; Do[k = ++c[DivisorSigma[0, j]]; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, Dec 12 2024 *)
%o A354606 (Python)
%o A354606 from sympy import divisor_count
%o A354606 from collections import Counter
%o A354606 def f(n): return divisor_count(n)
%o A354606 def aupton(nn):
%o A354606     an, fan, alst, inventory = 1, 1, [1], Counter([1])
%o A354606     for n in range(2, nn+1):
%o A354606         an = inventory[fan]
%o A354606         fan = f(an)
%o A354606         alst.append(an)
%o A354606         inventory.update([fan])
%o A354606     return alst
%o A354606 print(aupton(86)) # _Michael S. Branicky_, Jul 09 2022
%Y A354606 Cf. A355606, A000005, A124056, A342585.
%K A354606 nonn,easy
%O A354606 1,3
%A A354606 _Scott R. Shannon_, Jul 08 2022