

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
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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 distinctsized 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 raisedbit 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 64bit 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



