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A360706 a(n) is the least positive number not yet used such that its binary representation has either all or none of its 1-bits in common with the XOR of a(1) to a(n-1).

%I A360706 #51 Sep 01 2024 16:49:34
%S A360706 1,2,3,4,8,12,5,10,6,9,7,16,17,24,14,11,18,20,13,15,19,32,36,21,25,26,
%T A360706 22,23,27,33,37,28,34,30,29,40,42,31,64,96,35,68,38,44,41,43,39,48,56,
%U A360706 45,65,66,46,47,67,80,52,49,57,50,82,69,97,51,53,60,54,55,58,72,73,59,61,76,70,71,74
%N A360706 a(n) is the least positive number not yet used such that its binary representation has either all or none of its 1-bits in common with the XOR of a(1) to a(n-1).
%C A360706 The lexicographically earliest permutation of positive numbers such that the nim-sum of the first k elements equals the nim-sum of k-1 elements with the element at position k either arithmetically added or subtracted.
%C A360706 The first occurrence of a number m >= 2^k is always m = 2^k.
%C A360706 All positive integers will appear in this sequence: see link section for details.
%H A360706 Rémy Sigrist, <a href="/A360706/b360706.txt">Table of n, a(n) for n = 1..10000</a>
%H A360706 Thomas Scheuerle, <a href="/A360706/a360706.png">Scatterplot red: a(n) blue: a(1) XOR...XOR a(n-1) for n = 1...50000</a>
%H A360706 Thomas Scheuerle, <a href="/A360706/a360706.txt">Proof that all positive integers will appear in this sequence</a>
%H A360706 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H A360706 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A360706 If a(m1) = 2^k and a(m2) = 2^k-1 then m1 - 2^k < 0 and m2 - (2^k-1) > 0 for k > 2.
%e A360706    n    a(n)  a(n) in binary   a(1) XOR ... XOR a(n-1) in binary
%e A360706 ------------------------------------------------------------------
%e A360706    1     1          1b             0b
%e A360706    2     2         10b             1b
%e A360706    3     3         11b            11b
%e A360706    4     4        100b             0b
%e A360706    5     8       1000b           100b
%e A360706    6    12       1100b          1100b
%e A360706    7     5        101b             0b
%e A360706 ...
%e A360706 Signed version of this sequence such that the arithmetic sum over the first k values equals the nim-sum over the first k values of the original sequence:
%e A360706 1, 2, -3, 4, 8, -12, 5, 10, -6, -9, 7, 16, -17, 24, -14, 11, -18, 20, -13, ...
%o A360706 (MATLAB)
%o A360706 function a = A360706( max_n )
%o A360706     s = 0; a = []; t = [1:max_n];
%o A360706     for n = 1:max_n
%o A360706         k = 1;
%o A360706         while (t(k) ~= bitand(s,t(k)))&&(0 ~= bitand(s,t(k)))
%o A360706             k = k+1;
%o A360706         end
%o A360706         s = bitxor(s,t(k));
%o A360706         a(n) = t(k);
%o A360706         t(k) = max(t)+1; t = sort(t);
%o A360706     end
%o A360706 end
%o A360706 (PARI) { m = s = 0; for (n = 1, 77, for (v = 1, oo, if (!bittest(s, v), x = bitand(m, v); if (x==0 || x==v, s += 2^v; m = bitxor(m, v); print1 (v", "); break;);););); } \\ _Rémy Sigrist_, Aug 31 2024
%Y A360706 Cf. A000120, A116624, A269868, A330647, A360363, A375856.
%K A360706 nonn,base
%O A360706 1,2
%A A360706 _Thomas Scheuerle_, Feb 17 2023