A303769 a(0) = 0, a(n+1) is either the largest number obtained from a(n) by toggling a single 1-bit off (to 0) if no such number is yet in the sequence, otherwise the least number not yet in sequence that can be obtained from a(n) by toggling a single 0-bit on (to 1). In both cases the bit to be toggled is the rightmost possible that results yet an unencountered number.

0, 1, 3, 2, 6, 4, 5, 7, 15, 14, 12, 8, 9, 11, 10, 26, 24, 16, 17, 19, 18, 22, 20, 21, 23, 31, 30, 28, 29, 25, 27, 59, 58, 56, 48, 32, 33, 35, 34, 38, 36, 37, 39, 47, 46, 44, 40, 41, 43, 42, 106, 104, 96, 64, 65, 67, 66, 70, 68, 69, 71, 79, 78, 76, 72, 73, 75, 74, 90, 88, 80, 81, 83, 82, 86, 84, 85, 87, 95, 94, 92, 93, 89, 91, 123, 122, 120, 112, 113, 97, 99

Offset: 0

Author

Antti Karttunen, May 03 2018, with more direct definition from David A. Corneth, May 05 2018

Keywords

• nonn
• base

Comments

The original, but now conjectural, alternative definition is:

a(0) = 0 and for n > 0, if there are one or more k_i that are not already present in the sequence among terms a(0) .. a(n-1), and for which bitor(k_i,a(n-1)) = a(n-1), then a(n) = that k_i which gives maximal value of A019565(k_i) amongst them; otherwise, when no such k_i exists, a(n) = the least number not already present that can be obtained by toggling a single 0-bit of a(n-1) to 1. This is done by trying to toggle successive vacant bits from the least significant end of the binary representation of a(n-1), until such a sum a(n-1) + 2^h (= a(n-1) bitxor 2^h) is found that is not already present in the sequence.

The above construction is otherwise identical to that of A303767, except that we choose k_i with the maximal instead of minimal value of A019565.

In contrast to A303767, this sequence is not surjective (and thus not a permutation of nonnegative integers). The first missing term is 13 = A048675(70). See also comments in A303762, A303749 and A302775.

From David A. Corneth, May 05 2018: (Start)

Another description: a(0) = 0. a(n + 1) is the largest a(n) - 2^j > 0 that's not already in the sequence. If no such value exists, a(n + 1) is the least a(n) + 2^j not already in the sequence.

Using this definition we can prove that 13 isn't in the sequence. (End)

The equivalence of these definitions is still conjectural. The official definition of this sequence follows the latter one. - Antti Karttunen, Jun 08 2018

Links

• Antti Karttunen, Table of n, a(n) for n = 0..26232
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000120, A019565, A048675, A302774, A302775, A303749, A303762, A304749 (terms shown in base-2).

Cf. A303767 (a variant).

Programs

Mathematica

Nest[Append[#1, Min@ Select[{#2, #3, 2^IntegerLength[Last@ #1, 2] + Last@ #1}, IntegerQ]] & @@ Function[{a, d}, {a, SelectFirst[Sort@ Map[FromDigits[ReplacePart[d, First@ # -> 1], 2] &, Position[d, 0]], FreeQ[a, #] &], SelectFirst[Sort[#, Greater] &@ Map[FromDigits[ReplacePart[d, First@ # -> 0], 2] &, Position[d, 1]], FreeQ[a, #] &]}] @@ {#, IntegerDigits[Last@ #, 2]} &, {0}, 90] (* Michael De Vlieger, Jun 11 2018 *)

PARI

prepare_v303769(up_to) = { my(v = vector(up_to), occurred = Map(), prev=0, b); mapput(occurred,0,0); for(n=1,up_to, b=1; while(b<=prev, if(bitand(prev,b) && !mapisdefined(occurred,prev-b), v[n] = prev-b; break, b <<= 1)); if(!v[n], b=1; while(bitand(prev,b) || mapisdefined(occurred,prev+b), b <<= 1); v[n] = prev+b); mapput(occurred,prev = v[n],n)); (v); };
v303769 = prepare_v303769(16384);
A303769(n) = if(!n,n,v303769[n]); \\ Antti Karttunen, Jun 08 2018

Formula

a(n) = A048675(A303762(n)). [The original definition, now conjectured]

For n >= 0, A007088(a(n)) = A304749(n).

From David A. Corneth, May 05 2018: (Start)

The number of ones in the binary expansion of a(n) and a(n + 1) differ by 1. So A000120(a(n)) = A000120(a(n + 1)) +- 1. Furthermore, a(n + 1) <= 3 * a(n).

The number of binary digits of a(n + 1) is 0 or 1 more than the number of binary digits of a(n). So A070939(a(n + 1)) = A070939(a(n)) + 0 or 1. (End)