A341915 For any nonnegative number n with runs in binary expansion (r_1, ..., r_w), a(n) = Sum_{k = 1..w} 2^(r_1 + ... + r_k - 1).

0, 1, 3, 2, 5, 7, 6, 4, 9, 13, 15, 11, 10, 14, 12, 8, 17, 25, 29, 21, 23, 31, 27, 19, 18, 26, 30, 22, 20, 28, 24, 16, 33, 49, 57, 41, 45, 61, 53, 37, 39, 55, 63, 47, 43, 59, 51, 35, 34, 50, 58, 42, 46, 62, 54, 38, 36, 52, 60, 44, 40, 56, 48, 32, 65, 97, 113

Offset: 0

Author

Rémy Sigrist, Feb 23 2021

Keywords

• nonn
• look
• base

Comments

This sequence is a permutation of the nonnegative integers with inverse A341916.

This sequence has connections with A003188; here we compute partials sums of runs from left to right, there from right to left.

Examples

			For n = 23,
- the binary representation of 23 is "10111",
- the corresponding run lengths are (1, 1, 3),
- so a(23) = 2^(1-1) + 2^(1+1-1) + 2^(1+1+3-1) = 19.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8191
• Index entries for sequences related to binary expansion of n
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A003188, A005811, A059893, A101211, A341916 (inverse), A341943 (fixed points).

Programs

Mathematica

a[n_] := If[n == 0, 0, 2^((Length /@ Split[IntegerDigits[n, 2]] // Accumulate)-1) // Total];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 02 2022 *)

PARI

a(n) = { my (v=0); while (n, my (w=valuation(n+n%2,2)); n\=2^w; v=2^w*(1+v)); v/2 }

Formula

a(n) = A059893(A003188(n)).

a(n) = Sum_{k = 1..A005811(n)} 2^((Sum_{m = 1..k} A101211(m))-1).

a(n) < 2^k for any n < 2^k.

A000120(a(n)) = A000120(A003188(n)) = A005811(n).