A341915 - OEIS

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

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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).

If this sequence is grouped by the bit length of n into an irregular array with rows of lengths 1, 1, 2, 4, 8..., it becomes A088208 (resp. A362160) when each row is reversed and changed by a constant: a(n) = A088208(m, p-k) + p = A362160(m, p-k) + 2^m, where p = 2^m-1, m = floor(log_2(n)), k = n - 2^m, i.e., where n is the k-th number with m+1 bits in the binary expansion. - Andrei Zabolotskii, Dec 22 2025

EXAMPLE

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.

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 *)

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), A088208, A362160.