A330271 -id:A330271 - cp's OEIS frontend

A330270 a(n) is the least nonnegative integer k such that n XOR k is a square (where XOR denotes the bitwise XOR operator).

0, 0, 2, 2, 0, 1, 2, 3, 1, 0, 3, 2, 5, 4, 7, 6, 0, 1, 2, 3, 4, 5, 6, 7, 1, 0, 3, 2, 5, 4, 7, 6, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1

Offset: 0

Rémy Sigrist, Dec 08 2019

This sequence has similarities with A329794 as the XOR operator and the "box" operator defined in A329794 both map (n, n) to 0 for any n (however here we accept 0 as a square).

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

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 square, 0 <= x, y <= 1023

See A330271 for the cube variant.

Cf. A000290, A003987, A295520, A329794.

A330270[n_] := Module[{k = -1}, While[!IntegerQ[Sqrt[BitXor[n, ++k]]]]; k];
Array[A330270, 100, 0] (* Paolo Xausa, Feb 19 2024 *)

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

from itertools import count
from sympy.ntheory.primetest import is_square
def A330270(n): return next(k for k in count(0) if is_square(n^k)) # Chai Wah Wu, Aug 22 2023

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

a(a(n)) <= n.

A330272 a(n) is the least nonnegative integer k such that n OR k is a cube (where OR denotes the bitwise OR operator).

Original entry on oeis.org

0, 0, 25, 24, 121, 120, 337, 336, 0, 18, 17, 16, 113, 112, 3361, 3360, 11, 10, 9, 8, 105, 104, 321, 320, 3, 2, 1, 0, 97, 96, 29761, 29760, 93, 92, 1297, 1296, 89, 88, 3337, 3336, 85, 84, 3333, 3332, 81, 80, 3329, 3328, 77, 76, 1281, 1280, 73, 72, 59265, 59264

Offset: 0

Rémy Sigrist, Dec 08 2019

The sequence is well defined:
- for any k >= 0, the binary expansion of m = A000225(k)^3 has k trailing 1's,
- hence for any n < 2^k, n OR m = m, which is a cube, QED.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program for A330272
Rémy Sigrist, Scatterplot of (x, y) such that x OR y is a cube, 0 <= x, y <= 1023

See A330271 for the XOR variant.

Cf. A000225, A000578, A003986.

A330272[n_] := Module[{k = -1}, While[!IntegerQ[CubeRoot[BitOr[n, ++k]]]]; k];
Array[A330272, 60, 0] (* Paolo Xausa, Feb 20 2024 *)

PARI
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See Links section.
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a(n) = 0 iff n is a cube.

a(n) AND n = 0 (where AND denotes the bitwise AND operator).

cp's OEIS Frontend

A330270 a(n) is the least nonnegative integer k such that n XOR k is a square (where XOR denotes the bitwise XOR operator).

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