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

Author

Leroy Quet, Apr 30 2008

Keywords

• nonn
• base
• look
• easy

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

Crossrefs

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

Programs

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 *)

PARI

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

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)

Extensions

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