A330270 -id:A330270 - cp's OEIS frontend

A330271 a(n) is the least nonnegative integer k such that n XOR k is a cube (where XOR denotes the bitwise XOR operator).

0, 0, 2, 2, 4, 4, 6, 6, 0, 1, 2, 3, 4, 5, 6, 7, 11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4, 32, 32, 34, 34, 36, 36, 38, 38, 32, 33, 34, 35, 36, 37, 38, 39, 43, 42, 41, 40, 47, 46, 45, 44, 35, 34, 33, 32, 39, 38, 37, 36, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 0

Rémy Sigrist, Dec 08 2019

			For n = 4:
- 4 XOR 0 = 4 (not a cube),
- 4 XOR 1 = 5 (not a cube),
- 4 XOR 2 = 6 (not a cube),
- 4 XOR 3 = 7 (not a cube),
- 4 XOR 4 = 0 = 0^3,
- hence a(4) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of the ordinal transform of the first 2^20 terms
Rémy Sigrist, Scatterplot of (x, y) such that x XOR y is a cube, 0 <= x, y <= 1023

See A330270 for the square variant.
See A330272 for the OR variant.
Cf. A000578, A003987, A296615.

A330271[n_] := Module[{k = -1}, While[!IntegerQ[CubeRoot[BitXor[n, ++k]]]]; k];
Array[A330271, 100, 0] (* Paolo Xausa, Feb 20 2024 *)

a(n) = for (k=0, oo, if (ispower(bitxor(n,k),3), return (k)))

from itertools import count
from sympy import integer_nthroot
def A330271(n): return next(k for k in count(0) if integer_nthroot(n^k,3)[1]) # Chai Wah Wu, Aug 23 2023

a(n) = 0 iff n is a cube.

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

Original entry on oeis.org

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

Rémy Sigrist, May 12 2021

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.

			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.

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

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

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

a(n) = my (b); for (k=0, oo, if ((b=binary(bitxor(n, k)))==Vecrev(b), return (k)))

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

a(n) = 0 iff n belongs to A006995.
A070939(n XOR a(n)) = A070939(n).
A344259(n XOR a(n)) = A344259(n).

A331961 a(n) is the greatest square number k such that n AND k = k (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 1, 0, 1, 4, 4, 4, 4, 0, 9, 0, 9, 4, 9, 4, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, 16, 25, 16, 25, 16, 25, 0, 1, 0, 1, 36, 36, 36, 36, 0, 9, 0, 9, 36, 36, 36, 36, 16, 49, 16, 49, 36, 49, 36, 49, 16, 49, 16, 49, 36, 49, 36, 49, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Rémy Sigrist, Feb 02 2020

			The first terms, alongside the binary representations of n and of a(n), are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     0      10          0
   3     1      11          1
   4     4     100        100
   5     4     101        100
   6     4     110        100
   7     4     111        100
   8     0    1000          0
   9     9    1001       1001
  10     0    1010          0
  11     9    1011       1001
  12     4    1100        100
  13     9    1101       1001
  14     4    1110        100
  15     9    1111       1001
  16    16   10000      10000

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A062880, A330270.

a(n) = forstep (m=sqrtint(n), 0, -1, if (bitand(n, m^2)==m^2, return (m^2)))

from math import isqrt
def A331961(n): return next(m for m in (k**2 for k in range(isqrt(n),-1,-1)) if n&m==m) # Chai Wah Wu, Aug 22 2023

a(n) = 0 iff n belongs to A062880.

a(n^2) = n^2.

cp's OEIS Frontend

A330271 a(n) is the least nonnegative integer k such that n XOR k is a cube (where XOR denotes the bitwise XOR operator).

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

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A331961 a(n) is the greatest square number k such that n AND k = k (where AND denotes the bitwise AND operator).

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Python

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