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A359369 a(1) = 1. Thereafter a(n) = Sum_{j=1..n} {b(a(j)), where b(a(j)) = b(a(n))}, and b is A000120.

%I A359369 #33 Jul 03 2025 23:41:25
%S A359369 1,1,2,3,2,4,5,4,6,6,8,7,3,10,12,14,6,16,8,9,18,20,22,9,24,26,12,28,
%T A359369 15,4,10,30,8,11,18,32,12,34,36,38,21,24,40,42,27,12,44,30,16,13,33,
%U A359369 46,20,48,50,36,52,39,24,54,28,42,45,32,14,48,56,51,36,58,40,60,44,54,48,62,5
%N A359369 a(1) = 1. Thereafter a(n) = Sum_{j=1..n} {b(a(j)), where b(a(j)) = b(a(n))}, and b is A000120.
%C A359369 In other words, if k numbers having weight w have occurred, the most recent being a(n-1), then a(n) = k*w. Consequently every integer m > 1 appears A000005(m) times. Whilst there are > 2 ways composite number m may appear, there are only two ways for prime p. The first is consequent to the p_th occurrence of a power of 2. The final appearance of any number m is consequent to the first term in the sequence whose weight is m. For this reason final occurrences are very much delayed.
%C A359369 1 appears twice since A000120(1) = 1, the only fixed point in A000120.
%C A359369 First occurrences of primes are in natural order.
%C A359369 It appears that a(n) <= n, with equality at fixed points 1, 26, 28, ...
%C A359369 The plots have a curious net-like structure.
%C A359369 From _Michael De Vlieger_, Dec 29 2022: (Start)
%C A359369 a(186) = 188, and for n <= 2^20, there are 694462 occasions of a(n) > n.
%C A359369 Let w(n) = A000120(n) and let c_w(k) be the number of k in this sequence with binary weight w(k). Then this sequence consists of the recursive mapping of f(n) = w(a(n-1)) * c_w(a(n-1)).
%C A359369 Since f(n) is a product of 2 positive numbers, a(n) is odd iff both w(a(n-1)) and c_w(a(n-1)) are odd.
%C A359369 Let S_m = { k : w(k) = m }, thus, S_0 = {0}, S_1 = A000079, S_2 = A018900, S_3 = A014311, etc., with least element (2^m)-1 for m > 0.
%C A359369 Let trajectory T_m comprise a(j) such that w(a(j)) = m. Then a(j) is in S_m.
%C A359369 If the a(j) in T_m appear in order of j, then T_m(1) is such that c_m = 1, T_m(2) is such that c_m = 2, and generally c_m(T_m(k)) = k.
%C A359369 This sequence is composed of trajectories T_m evident in scatterplot. (End)
%H A359369 Michael De Vlieger, <a href="/A359369/b359369.txt">Table of n, a(n) for n = 1..16384</a>
%H A359369 Michael De Vlieger, <a href="/A359369/a359369.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A359369 Michael De Vlieger, <a href="/A359369/a359369_1.png">Log log scatterplot of a(n)</a> n = 1..2^16, with a color function showing m = A000120(a(n-1)) with m = 1 in red, m = 2 in orange, ..., m = 15 in magenta.
%e A359369 1 appears twice only, first as given starting term a(1), then as a(2) = 1.
%e A359369 a(7) = 5, consequent to a(6)=4, the 5th term so far with binary weight = 1.
%e A359369 a(77) = 5, consequent to a(76) = 62, the first occurrence of a term with binary weight = 5. These are the only occurrences of 5 in the sequence.
%e A359369 The first occurrence of 8 is a(11), following a(10) = 6, the fourth term with weight 2.
%e A359369 a(11) = 8 is the 7th term with weight 1, and a(12) is the first occurrence of 7.
%t A359369 Block[{a, c, f, k, nn}, nn = 76; c[_] = 0; a[1] = 1; f[n_] := DigitCount[n, 2, 1]; Do[Set[k, ( c[#]++; # c[#]) &[f[#]]] &@a[n - 1]; Set[a[n], k], {n, 2, nn}]; Array[a, nn] ] (* _Michael De Vlieger_, Dec 28 2022 *)
%Y A359369 Cf. A000005, A000120, A000225, A359384.
%K A359369 nonn,look
%O A359369 1,3
%A A359369 _David James Sycamore_, Dec 28 2022