A344220 a(n) is the least k >= 0 such that n XOR k is a binary palindrome (where XOR denotes the bitwise XOR operator).

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 2, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 3, 2, 3, 2, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 5, 4, 7, 6, 5, 4, 7, 6, 1, 0, 3, 2, 3, 2, 1, 0, 7, 6, 5, 4, 7, 6, 5, 4, 3, 2, 1, 0, 1, 0, 3, 2, 5, 4, 7, 6, 1, 0, 3, 2, 5, 4, 7, 6, 5, 4, 7, 6, 1, 0, 3

Offset

0,11

Comments

The binary expansions of n and of n XOR a(n) have the same length, say k, and their first ceiling(k/2) bits are the same.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Rémy Sigrist, Scatterplot of the ordinal transform of the first 2^18 terms

Rémy Sigrist, Scatterplot of (x, y) such that x XOR y is a binary palindrome and x, y < 1024

Formula

a(n) = 0 iff n belongs to A006995.

A070939(n XOR a(n)) = A070939(n).

A344259(n XOR a(n)) = A344259(n).

Example

For n=42:
- 42 XOR 0 = 42 ("101010" in binary) is not a binary palindrome,
- 42 XOR 1 = 43 ("101011" in binary) is not a binary palindrome,
- 42 XOR 2 = 40 ("101000" in binary) is not a binary palindrome,
- 42 XOR 3 = 41 ("101001" in binary) is not a binary palindrome,
- 42 XOR 4 = 46 ("101110" in binary) is not a binary palindrome,
- 42 XOR 5 = 47 ("101111" in binary) is not a binary palindrome,
- 42 XOR 6 = 44 ("101100" in binary) is not a binary palindrome,
- 42 XOR 7 = 45 ("101101" in binary) is a binary palindrome,
- so a(42) = 7.

Mathematica

A344220[n_] := Module[{k = -1}, While[!PalindromeQ[IntegerDigits[BitXor[n, ++k], 2]]]; k]; Array[A344220, 100, 0] (* Paolo Xausa, Feb 19 2024 *)

Prog

(PARI) a(n) = my (b); for (k=0, oo, if ((b=binary(bitxor(n, k)))==Vecrev(b), return (k)))
(Python)
from itertools import count
def A344220(n): return next(k for k in count(0) if (s := bin(n^k)[2:])[:(t:=len(s)+1>>1)]==s[:-t-1:-1]) # Chai Wah Wu, Aug 23 2023

Crossrefs

Cf. A006995, A070939, A295520, A330270, A344259.

Sequence in context: A307551 A220344 A176102 * A318056 A096839 A099891

Adjacent sequences: A344217 A344218 A344219 * A344221 A344222 A344223

Keyword

nonn,base

Author

Rémy Sigrist, May 12 2021

Status

approved