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A343327 Start with a(1)=1; thereafter a(n+1) = Sum_{k=1..n such that d(a(k)) = d(a(n))} d(a(k)) where d = A000005.

%I A343327 #51 Dec 10 2023 17:22:56
%S A343327 1,1,2,2,4,3,6,4,6,8,12,6,16,5,8,20,12,18,24,8,24,16,10,28,30,24,32,
%T A343327 36,9,9,12,42,40,48,10,32,48,20,54,56,64,7,10,36,18,60,12,66,72,24,80,
%U A343327 30,88,96,36,27,40,104,112,40,120,16,15,44,72,48,50,78,128,136,144,15
%N A343327 Start with a(1)=1; thereafter a(n+1) = Sum_{k=1..n such that d(a(k)) = d(a(n))} d(a(k)) where d = A000005.
%C A343327 Every nonzero number occurs in the sequence since all terms in A000005 are included. The multiplicity of any term m (> 2) is d(m)-1 (since m > 2 cannot be created by adding m 1s). Prime 2 occurs twice, consecutively (see example), and thereafter no two adjacent terms can be prime because the term prior to a prime q is the first occurrence of 2^(q-1) (composite for q > 2), and since d(q)=2, the following term is a multiple (> 1) of 2. Thus each prime > 2 is flanked by composite even numbers, and occurs once only. Conjecture: The primes, after the first appearance of 2, appear in their natural order.
%C A343327 The primes appear in order at a(n) for n in {3, 4, 6, 14, 42, 507, 1939, 22454, 90322, ...}, with a(3) = a(4) = 2. No further primes appear for n <= 2^20. - _Michael De Vlieger_, May 01 2021
%H A343327 Michael De Vlieger, <a href="/A343327/b343327.txt">Table of n, a(n) for n = 1..10000</a>
%H A343327 Michael De Vlieger, <a href="/A343327/a343327.png">Scatterplot of a(n)</a> for n = 1..2^20 showing trajectories of tau(a(n)).
%H A343327 Michael De Vlieger, <a href="/A343327/a343327_1.png">Annotated scatterplot of a(n)</a> for n = 1..256, noting and illustrating tau(a(n)) via color function.
%e A343327 Given a(1)=1, a(2) is also 1, since a(1) has no prior term. a(3)=1+1=2 and a(4)=2 because a(3) has 2 divisors and there is no prior term with 2 divisors. a(5)=2+2=4.
%t A343327 Block[{a = {1}, c}, Do[(If[! IntegerQ[c[#]], Set[c[#], 1], c[#]++]; AppendTo[a, # c[#]]) &@ DivisorSigma[0, a[[-1]]], 71]; a] (* _Michael De Vlieger_, Apr 12 2021 *)
%o A343327 (PARI) lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = sum(k=1, n-1, if (numdiv(va[k]) == numdiv(va[n-1]), numdiv(va[k])));); va;} \\ _Michel Marcus_, Apr 28 2021
%Y A343327 Cf. A000005, A009087, A005179, A061286.
%K A343327 nonn
%O A343327 1,3
%A A343327 _David James Sycamore_, Apr 12 2021