A359384 a(1) = 0. If a(n-1) is a first occurrence, a(n) = A000120(a(n-1)). Otherwise, if a(n-1) is a repeat of a prior terms, a(n) = number of indices j < n such that a(j) = a(n-1).

0, 0, 2, 1, 1, 2, 2, 3, 2, 4, 1, 3, 2, 5, 2, 6, 2, 7, 3, 3, 4, 2, 8, 1, 4, 3, 5, 2, 9, 2, 10, 2, 11, 3, 6, 2, 12, 2, 13, 3, 7, 2, 14, 3, 8, 2, 15, 4, 4, 5, 3, 9, 2, 16, 1, 5, 4, 6, 3, 10, 2, 17, 2, 18, 2, 19, 3, 11, 2, 20, 2, 21, 3, 12, 2, 22, 3, 13, 2, 23, 4

Offset: 1

Author

David James Sycamore, Dec 27 2022

Keywords

• nonn

Comments

In other words, a novel term is followed by its binary weight and a repeat term is followed by its cardinality. The sequence is infinite, and all nonzero numbers appear infinitely many times. Zero occurs just twice because it is the only number with zero binary weight. 1 occurs following powers of 2.

a(n) < n.

Examples

			a(8) = 3, a novel term, therefore a(9) = 2, the binary weight of 3.
a(12) = 3, occurring for the 2nd time, so a(13) = 2.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 3..2^20.
• Michael De Vlieger, Log log scatterplot of a(n), n = 3..2^16, with a color function showing m = A000120(a(n-1)) in black for m = 0, red for m = 1, orange for m = 2, ..., magenta for m = 12.

Crossrefs

Cf. A000120, A359369.

Programs

Mathematica

Block[{a, c, f, k, nn}, nn = 80; c[] = 0; a[1] = 0; f[n] := DigitCount[n, 2, 1]; Do[If[c[#] == 0, c[#]++; Set[k, f[#]], c[#]++; Set[k, c[#]]] &[a[n - 1]]; a[n] = k, {n, 2, nn}]; Array[a, nn] ] (* Michael De Vlieger, Dec 28 2022 *)