A268415 Balanced odious numbers: numbers with an odd number of runs of 1's in their binary expansion.

1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 21, 24, 28, 30, 31, 32, 37, 41, 42, 43, 45, 48, 53, 56, 60, 62, 63, 64, 69, 73, 74, 75, 77, 81, 82, 83, 84, 86, 87, 89, 90, 91, 93, 96, 101, 105, 106, 107, 109, 112, 117, 120, 124, 126, 127, 128, 133, 137, 138, 139, 141

Offset: 1

Author

Vladimir Shevelev, Feb 04 2016

Keywords

• nonn
• base

Comments

In the balanced binary system the sequence A268411 plays the role of the Thue-Morse sequence (A010060). Therefore we call the balanced odious numbers those numbers n for which A268411(n) = 1.

Complement of A268412.

Examples

			77 is a member because its binary expansion (1001101) has 3 runs of 1's, and 3 is odd.
		

Links

• Peter J. C. Moses (terms 1..1000) & Antti Karttunen, Table of n, a(n) for n = 1..8129
• Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Crossrefs

Positions of odd terms in A069010.

Cf. A010060, A000069, A268411.

Cf. A268412 (complement).

Cf. A268382 (the least monotonic left inverse).

Cf. A268477 (primes in this sequence).

Programs

Mathematica

balancedBinary:=Join[#,{0}]-Join[{0},#]&[IntegerDigits[#,2]]&;
Flatten[Position[Map[Mod[Count[balancedBinary[#],1],2]&,Range[0,100]],1]-1] (* Peter J. C. Moses, Feb 04 2016 *)
Select[Range[200],OddQ[Count[Split[IntegerDigits[#,2]],?(MemberQ[ #,1]&)]]&] (* _Harvey P. Dale, Mar 31 2019 *)

Python

A268415_list = [i for i in range(10**6) if len(list(filter(bool,format(i,'b').split('0')))) % 2] # Chai Wah Wu, Mar 01 2016

Scheme

(define A268415 (ZERO-POS 1 1 (COMPOSE -1+ A268411))) ;; requires also my IntSeq-library. - Antti Karttunen, Feb 05 2016

Formula

Other identities. For all n >= 1:

A268382(a(n)) = n.

Extensions

More terms from Peter J. C. Moses, Feb 04 2016