OFFSET
1,3
COMMENTS
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.
First occurrences of primes are in natural order.
It appears that a(n) <= n, with equality at fixed points 1, 26, 28, ...
The plots have a curious net-like structure.
From Michael De Vlieger, Dec 29 2022: (Start)
a(186) = 188, and for n <= 2^20, there are 694462 occasions of a(n) > n.
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)).
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.
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.
Let trajectory T_m comprise a(j) such that w(a(j)) = m. Then a(j) is in S_m.
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.
This sequence is composed of trajectories T_m evident in scatterplot. (End)
LINKS
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n) 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.
EXAMPLE
1 appears twice only, first as given starting term a(1), then as a(2) = 1.
a(7) = 5, consequent to a(6)=4, the 5th term so far with binary weight = 1.
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.
The first occurrence of 8 is a(11), following a(10) = 6, the fourth term with weight 2.
a(11) = 8 is the 7th term with weight 1, and a(12) is the first occurrence of 7.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, Dec 28 2022
STATUS
approved