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

1,2

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.

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.

Formula

Other identities. For all n >= 1:

A268382(a(n)) = n.

Example

Since in the binary 77=1001101 has odd number three) runs of 1's, then 77 is a member.

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 *)

Prog

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

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).

Sequence in context: A191281 A032900 A370684 * A352084 A277779 A166935

Adjacent sequences: A268412 A268413 A268414 * A268416 A268417 A268418

Keyword

nonn,base

Author

Vladimir Shevelev, Feb 04 2016

Extensions

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

Status

approved