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A339793 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that is a multiple of s(a(n-1)), the sum of the proper divisors of a(n-1).

%I A339793 #29 Jan 16 2023 09:10:46
%S A339793 1,2,3,4,6,12,16,15,9,8,7,5,10,24,36,55,17,11,13,14,20,22,28,56,64,63,
%T A339793 41,18,21,33,30,42,54,66,78,90,144,259,45,99,57,23,19,25,48,76,128,
%U A339793 127,26,32,31,27,39,34,40,50,43,29,35,52,46,104,106,112,136,134,70,74,80,212,166,86,92,152
%N A339793 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that is a multiple of s(a(n-1)), the sum of the proper divisors of a(n-1).
%C A339793 The sequence is possibly a permutation of the positive integers as when a(n-1) is prime a(n) will be the next smallest number that has not previously occurred. However this will depend on the likelihood of a(n) being a prime as n goes to infinity. For the first 478 terms the last prime is a(144) = 59, while a(478) = 19140499834691254267668, indicating prime values become increasingly rare, and could potentially have a finite number as n->infinity.
%C A339793 The sum of the proper divisors of n is given by A001065(n).
%H A339793 Scott R. Shannon, <a href="/A339793/b339793.txt">Table of n, a(n) for n = 1..478</a>
%H A339793 Wikipedia, <a href="https://en.wikipedia.org/wiki/Aliquot_sum">Aliquot sum</a>.
%e A339793 a(3) = 3 as s(a(2)) = s(2) = 1, and 3 is the smallest multiple of 1 that has not previously occurred.
%e A339793 a(5) = 6 as s(a(4)) = s(4) = 3, and as 3 has already occurred the next lowest multiple is used, being 6.
%e A339793 a(12) = 5 as s(a(11)) = s(7) = 1, and 5 is the smallest multiple of 1 that has not previously occurred.
%o A339793 (Python)
%o A339793 from sympy import divisors
%o A339793 def s(k): return sum(d for d in divisors(k)[:-1])
%o A339793 def aupto(n):
%o A339793   alst, aset = [1, 2], {1, 2}
%o A339793   for k in range(2, n):
%o A339793     ak = sanm1 = s(alst[-1])
%o A339793     while ak in aset: ak += sanm1
%o A339793     alst.append(ak); aset.add(ak)
%o A339793   return alst     # use alst[n-1] for a(n)
%o A339793 print(aupto(478)) # _Michael S. Branicky_, Dec 29 2020
%Y A339793 Cf. A001065, A027751, A000203, A032741, A032742, A007956.
%K A339793 nonn
%O A339793 1,2
%A A339793 _Scott R. Shannon_, Dec 17 2020