

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).


4



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
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OFFSET

0,3


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.


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


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).


EXAMPLE

For n = 23,
 the binary representation of 23 is "10111",
 the corresponding run lengths are (1, 1, 3),
 so a(23) = 2^(11) + 2^(1+11) + 2^(1+1+31) = 19.


MATHEMATICA

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


PROG

(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 }


CROSSREFS

Cf. A003188, A005811, A059893, A101211, A341916 (inverse), A341943 (fixed points).
Sequence in context: A108918 A316472 A341910 * A082334 A294371 A325985
Adjacent sequences: A341912 A341913 A341914 * A341916 A341917 A341918


KEYWORD

nonn,look,base


AUTHOR

Rémy Sigrist, Feb 23 2021


STATUS

approved



