A341943 -id:A341943 - cp's OEIS frontend

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

Rémy Sigrist, Feb 23 2021

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.

			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.

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

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 *)

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

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).

A341916 Inverse permutation to A341915.

Original entry on oeis.org

0, 1, 3, 2, 7, 4, 6, 5, 15, 8, 12, 11, 14, 9, 13, 10, 31, 16, 24, 23, 28, 19, 27, 20, 30, 17, 25, 22, 29, 18, 26, 21, 63, 32, 48, 47, 56, 39, 55, 40, 60, 35, 51, 44, 59, 36, 52, 43, 62, 33, 49, 46, 57, 38, 54, 41, 61, 34, 50, 45, 58, 37, 53, 42, 127, 64, 96

Offset: 0

Rémy Sigrist, Feb 24 2021

			A341915(5) = 7, so a(7) = 5.

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
Rémy Sigrist, PARI program for A341916

Cf. A006068, A059893, A341915 (inverse), A341943 (fixed points).

PARI
```
See Links section.
```

a(n) = A006068(A059893(n)).

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

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula