A139708 Take n in binary. Rotate the binary digits to the left until a 1 once again appears as the leftmost digit. Convert back into decimal for a(n).

1, 2, 3, 4, 6, 5, 7, 8, 12, 10, 14, 9, 11, 13, 15, 16, 24, 20, 28, 18, 22, 26, 30, 17, 19, 21, 23, 25, 27, 29, 31, 32, 48, 40, 56, 36, 44, 52, 60, 34, 38, 42, 46, 50, 54, 58, 62, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 96, 80, 112, 72, 88, 104, 120

Offset

1,2

Comments

This sequence written in binary is A139709.

This is a permutation of the positive integers. A139706 is the inverse permutation.

Moreover, the first 2^n terms are a permutation of the first 2^n positive integers. Fixed points of the permutation are A272919. - Ivan Neretin, May 10 2016

References

Lionel Levine, Fractal sequences and restricted Nim, Ars Combin. 80 (2006), 113-127.

Links

Ivan Neretin, Table of n, a(n) for n = 1..8192

Lionel Levine, Fractal sequences and restricted Nim, arXiv:math/0409408 [math.CO], 2004.

Index entries for sequences that are permutations of the natural numbers

Formula

From Mikhail Kurkov, Dec 23 2023: (Start)

a(2^m + k) = f(2^m + f(k)) for m >= 0, 0 <= k < 2^m where f(n) = A059893(n) for n > 0 with f(0) = 0.

a(n) = f(A139706(f(n))). (End)

Maple

A139708 := proc(n) local a; a := ListTools[Rotate](convert(n, base, 2), -1) ; while op(-1, a) = 0 do a := ListTools[Rotate](a, -1) ; od: add(op(i, a)*2^(i-1), i=1..nops(a)) : end: seq(A139708(n), n=1..100) ; # R. J. Mathar, May 04 2008

Mathematica

rbd[n_]:=Module[{idn2=RotateLeft[IntegerDigits[n, 2]]}, While[ idn2[[1]] ==0, idn2= RotateLeft[ idn2]]; FromDigits[idn2, 2]]; Array[rbd, 80] (* Harvey P. Dale, Jun 07 2015 *)
Table[FromDigits[RotateLeft[d = IntegerDigits[n, 2], Position[Join[d, d], 1][[2, 1]] - 1], 2], {n, 71}] (* Ivan Neretin, May 10 2016 *)

Prog

(PARI) a(n) = if(bitand(n, n-1)==0, n, my(b=logint(n, 2), s=b-logint(n-(1<<b), 2)); (((n - (1<<b)) << 1) + 1) << (s-1)) \\ Andrew Howroyd, Jan 04 2024

Crossrefs

Cf. A139706 (inverse), A139709 (in binary), A272919 (fixed points).

Sequence in context: A139706 A333777 A306869 * A333776 A305410 A059893

Adjacent sequences: A139705 A139706 A139707 * A139709 A139710 A139711

Keyword

nonn,base,look,easy

Author

Leroy Quet, Apr 30 2008

Extensions

More terms from R. J. Mathar, May 04 2008

Status

approved