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A354960 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that is a multiple of the number of proper divisors of a(n-1).

%I A354960 #47 Jan 16 2023 09:10:46
%S A354960 1,2,3,4,6,9,8,12,5,7,10,15,18,20,25,14,21,24,28,30,35,27,33,36,16,32,
%T A354960 40,42,49,22,39,45,50,55,48,54,56,63,60,11,13,17,19,23,26,51,57,66,70,
%U A354960 77,69,72,44,65,75,80,81,52,85,78,84,88,91,87,90,99,95,93,96,110,98,100,64,102,105,112
%N A354960 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that is a multiple of the number of proper divisors of a(n-1).
%C A354960 The terms are concentrated along numerous lines, some of which curve upward while others curve downward. See the first linked image. Surprisingly these lines are not shared by terms which are a multiple of a given proper divisor count, but predominantly by terms sharing a certain prime factor. See the second linked image.
%C A354960 The sequence is conjectured to be a permutation of the positive integers although it may take an extremely large number of terms for the primes to appear; e.g., 263 has not occurred after 500000 terms. Also although the vast majority of primes will appear in their natural order, some may not; e.g., a(455) = 840, which has 31 proper divisors, so a(456) = 31, and then a(457) = 29.
%C A354960 In the first 500000 terms the only fixed points beyond the first two are 3, 4, 1159, 1207. It is possible that no more exist, although this is unknown.
%H A354960 Michael De Vlieger, <a href="/A354960/b354960.txt">Table of n, a(n) for n = 1..10000</a>
%H A354960 Michael De Vlieger, <a href="/A354960/a354960_3.png">Annotated log-log scatterplot of a(n)</a>, n - 1..2^16, showing records in red, local minima in blue, fixed points in gold, primes in green, and prime powers in cyan.
%H A354960 Michael De Vlieger, <a href="/A354960/a354960_4.png">Log-log scatterplot of a(n)</a> n = 1..2^12, with a color function indicating tau(a(n-1))-1.
%H A354960 Scott R. Shannon, <a href="/A354960/a354960.png">Image of the first 50000 terms</a>. The green line is y = n.
%H A354960 Scott R. Shannon, <a href="/A354960/a354960_2.png">Image of the first 50000 terms with color</a>. Terms containing prime factors 17, 13, 11, 7, 5, 3, 2 are colored purple, blue, green, yellow, orange, red, and white respectively. All other terms are colored gray.
%H A354960 Scott R. Shannon, <a href="/A354960/a354960_1.png">Image of the first 500000 terms</a>.
%e A354960 a(3) = 3 as a(2) = 2 which has one proper divisor, and 2 is the smallest unused multiple of 1.
%e A354960 a(5) = 6 as a(4) = 4 which has two proper divisors, and 6 is the smallest unused multiple of 2.
%e A354960 a(9) = 5 as a(8) = 12 which has five proper divisors, and 5 is the smallest unused multiple of 5.
%t A354960 nn = 76; c[_] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = 2; Do[k = DivisorSigma[0, a[n - 1]] - 1; m = 1 + Floor[u/k]; While[c[m k], m++]; a[n] = m k; c[m k] = True; If[a[n] == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger_, Jul 24 2022 *)
%Y A354960 Cf. A128556 (all divisors), A339793, A000120, A032741, A005179, A027751, A335372.
%K A354960 nonn,look
%O A354960 1,2
%A A354960 _Scott R. Shannon_, Jul 23 2022