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.

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, 15, 4, 10, 30, 8, 11, 18, 32, 12, 34, 36, 38, 21, 24, 40, 42, 27, 12, 44, 30, 16, 13, 33, 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

Offset: 1

Author

David James Sycamore, Dec 28 2022

Keywords

• nonn
• look

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.

1 appears twice since A000120(1) = 1, the only fixed point in A000120.

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)

Examples

			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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• 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.

Crossrefs

Cf. A000005, A000120, A000225, A359384.

Programs

Mathematica

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 *)