%I #36 May 29 2024 07:04:32
%S 0,1,2,3,3,2,1,0,4,5,6,7,7,6,5,4,5,4,7,6,6,7,4,5,1,0,3,2,2,3,0,1,6,7,
%T 4,5,5,4,7,6,2,3,0,1,1,0,3,2,3,2,1,0,0,1,2,3,7,6,5,4,4,5,6,7,7,6,5,4,
%U 4,5,6,7,3,2,1,0,0,1,2,3,2,3,0,1,1,0
%N a(n) = bitwise XOR of all the bit numbers for the bits that are set in n, using number 1 for the LSB.
%C If the least significant bit is numbered 0, then a(2n) = a(2n+1) if one uses the "natural" definition reading "...set in n": see A253315 for that version. To avoid the duplication, we chose here to start numbering the bits with 1 for the LSB; equivalently, we can start numbering the bits with 0 but use the definition "...bits set in 2n". In any case, a(n) = A253315(2n) = A253315(2n+1).
%C Since the XOR operation is associative, one can define XOR of an arbitrary number of terms in a recursive way, there is no ambiguity about the order in which the operations are performed.
%H Philippe Beaudoin, <a href="/A261283/b261283.txt">Table of n, a(n) for n = 0..8190</a>
%H Oliver Nash, <a href="http://olivernash.org/2009/10/31/yet-another-prisoner-puzzle/index.html">Yet another prisoner puzzle</a>, coins on a chessboard problem.
%H Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:OR_and_XOR_of_binary_exponents.svg">Illustration of the first 128 terms</a>
%e a(7) = a(4+2+1) = a(2^2+2^1+2^0) = (2+1) XOR (1+1) XOR (0+1) = 3 XOR 3 = 0.
%e a(12) = a(8+4) = a(2^3+2^2) = (3+1) XOR (2+1) = 4+3 = 7.
%t A261283[n_] := If[n == 0, 0, BitXor @@ PositionIndex[Reverse[IntegerDigits[n, 2]]][1]]; Array[A261283, 100, 0] (* _Paolo Xausa_, May 29 2024 *)
%o (PARI) A261283(n,b=bittest(n,0))={for(i=1,#binary(n),bittest(n,i)&&b=bitxor(b,i+1));b}
%Y Cf. A075926 (indices of 0's), A253315, A327041 (OR equivalent).
%K nonn,base,easy
%O 0,3
%A _M. F. Hasler_, Aug 14 2015, following the original version A253315 by _Philippe Beaudoin_, Dec 30 2014