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A341242 Numbers whose binary representation encodes a subset S of the natural numbers such that the XOR of the binary representations of all s in S gives 0. 0
0, 1, 14, 15, 50, 51, 60, 61, 84, 85, 90, 91, 102, 103, 104, 105, 150, 151, 152, 153, 164, 165, 170, 171, 194, 195, 204, 205, 240, 241, 254, 255, 770, 771, 780, 781, 816, 817, 830, 831, 854, 855, 856, 857, 868, 869, 874, 875, 916, 917, 922, 923, 934, 935, 936 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The numbers for which the set S of positions of bits 1 in the binary representation, interpreted as a set of distinct-sized Nim heaps (including a possible uninteresting size 0 heap for the least significant bit) is losing for the player to move.

Viewing the list as a set of valid code words, every natural number N can be "corrected" to a valid code word by changing exactly one bit, in exactly one way. The position of that bit is found by computing for N the XOR of its raised-bit positions of the title (if the result is 0, then N is already valid but flipping the irrelevant bit 0 makes it valid again).

The "error correcting" interpretation, applied to 64-bit numbers interpreted as orientation of 64 coins, corresponds to a solution of the "coins on a chessboard" puzzle described in the Nick Berry's blog, and also mentioned at A253315.

Numbers 2*n and 2*n+1 for n = A075926(m).

Numbers m such that A253315(m) = 0. - Rémy Sigrist, Feb 09 2021

LINKS

Table of n, a(n) for n=1..55.

Nick Berry, Impossible Escape?, DataGenetics blog, December 2014.

FORMULA

a(2*n+1) = 2*A075926(n), a(2*n+2) = 2*A075926(n) + 1 for any n >= 0. - Rémy Sigrist, Feb 09 2021

PROG

(C++) (first 2^12 terms)

for (int i=0; i<65536; ++i)  {

    int sum=0;

    for (int n=i, count=0; n>0; n>>=1, ++count)

      if ((n&1)!=0)

        sum ^= count;

    if (sum==0)

      std::cout << i << ", ";

  }

(Python)

def ok(n):

  xor, b = 0, (bin(n)[2:])[::-1]

  for i, c in enumerate(b):

    if c == '1': xor ^= i

  return xor == 0

print([m for m in range(937) if ok(m)]) # Michael S. Branicky, Feb 07 2021

CROSSREFS

Cf. A075926, A253315.

Sequence in context: A041404 A041402 A041929 * A174924 A041408 A041406

Adjacent sequences:  A341239 A341240 A341241 * A341243 A341244 A341245

KEYWORD

nonn,base

AUTHOR

Marc A. A. van Leeuwen, Feb 07 2021

STATUS

approved

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Last modified September 25 09:40 EDT 2021. Contains 347654 sequences. (Running on oeis4.)