A322463 Reverse runs of zeros in binary expansion of n and convert back to decimal.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 19, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 40, 35, 36, 41, 44, 39, 34, 37, 42, 43, 38, 45, 46, 47, 48, 49, 52, 51, 50, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 80, 67

Offset: 0

Author

Rémy Sigrist, Dec 09 2018

Keywords

• nonn
• base
• look

Comments

This sequence is a self-inverse permutation of nonnegative integers.

Examples

			For n = 150:
- the binary representation of 150 is "10010110",
- we have three runs of zeros: "00", "0" and "0",
- we exchange the first and the third run, and the second remains in place,
- we obtain: "10101100",
- hence a(150) = 172.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 0..2^18-1 (where the color is function of A005811(n))
• Index entries for sequences related to binary expansion of n
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

See A322464 for the variant where we reverse the runs of ones.

See A056539 for a similar sequence.

Cf. A000120, A005811, A164710.

Programs

Mathematica

a[n_] := Module[{s=Split[IntegerDigits[n,2]]}, m=Length[s]; m2=m-Mod[m,2]; If[m2>0, ind=Riffle[Range[1,m,2],Range[m2,1,-2]]; FromDigits[Flatten[s[[ind]]],2],n]]; Array[a, 100, 0] (* Amiram Eldar, Dec 12 2018 *)

PARI

a(n) = {
    my (r=n, z=[], v=0, p=1, i=0);
    while (r, my (l=valuation(r+(r%2),2)); if (r%2==0, z=concat(l,z)); r\=2^l);
    while (n, my (l=valuation(n+(n%2),2)); if (n%2, v+=(2^l-1)*p; p*=2^l, p*=2^z[i++]); n\=2^l);
    return (v);
}

Formula

A000120(a(n)) = A000120(n).

A005811(a(n)) = A005811(n).

a(A164710(n)) = A164710(n).

a(A322464(n)) = A322464(a(n)).

a(2^n) = 2^n.

a(2^n-1) = 2^n-1.