A043529 Number of distinct base-2 digits of n.

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Also, if prefixed by 0, the trajectory of 0 under repeated applications of the morphism 0 -> 0,1, 1 -> 1,2, 2 -> 2,2. This is a word that is pure uniform morphic, but neither primitive morphic nor recurrent. - N. J. A. Sloane, Jul 15 2018

References

• Dekking, Michel, Michel Mendes France, and Alf van der Poorten. "Folds." The Mathematical Intelligencer, 4.3 (1982): 130-138 & front cover, and 4:4 (1982): 173-181 (printed in two parts). See Observaion 1.8.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16384
• Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017. See Example 35.
• Index entries for sequences related to binary expansion of n

Crossrefs

Factor of A160466. Cf. A007456 and A081729. - Johannes W. Meijer, May 24 2009

Cf. A007088, A036987, A043545.

Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

Programs

Maple

A043529 := proc(n): if type(ln(n+1)/ln(2), integer) then 1 else 2 fi: end proc: seq(A043529(n), n=0..90); # Johannes W. Meijer, Sep 14 2012

Mathematica

(* Needs version >= 10.2. *)
SubstitutionSystem[{0 -> {0, 1}, 1 -> {1, 2}, 2 -> {2, 2}}, 0, 7] // Last // Rest (* Jean-François Alcover, Apr 06 2020 *)
Table[Length[Union[IntegerDigits[n,2]]],{n,0,90}] (* Harvey P. Dale, Aug 04 2024 *)

Formula

This is 2 unless n = 2^k - 1 for some k in which case it is 1.

a(n) = 2 - A036987(n). - Antti Karttunen, Nov 19 2017

Extensions

First term added and offset changed by Johannes W. Meijer, May 15 2009