A329794 -id:A329794 - 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.

A329795 a(n) = smallest positive k such that scan_diff(k,n) is a square, where scan_diff is defined in the Comments.

Original entry on oeis.org

2, 1, 2, 3, 1, 2, 3, 4, 5, 3, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 1, 2, 1, 2, 3, 4, 5, 1, 2, 3, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 1, 2, 1, 2, 3, 4, 5, 6

Offset: 1

N. J. A. Sloane, Dec 07 2019

Write i, j in base 10 aligned to right, say
i = bcd...ef
j = .gh...pq
Then scan_diff(i,j) = |b-0| + |c-g| + |d-h| + ... + |e-p| + |f-q|.
Example: scan_diff(12345,909) = 1+2+6+4+4 = 17.
Suggested by the definition of "box" in A329794.

			For n = 1 the smallest k producing a square is 2 (as scan_diff(1,2) = 1);
For n = 2 the smallest k producing a square is 1 (as scan_diff(2,1) = 1);
For n = 3 the smallest k producing a square is 2 (as scan_diff(3,2) = 1);
For n = 5 the smallest k producing a square is 1 (as scan_diff(5,1) = 4);
For n = 16 the smallest k producing a square is 3 (as scan_diff(16,3) = 1+3 = 4).

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Cf. A329794.

scan_diff(n,k) = if (n*k, scan_diff(n\10,k\10)+abs((n%10)-(k%10)), n+k)
a(n) = for (k=1, oo, my (t=scan_diff(n,k)); if (t && issquare(t), return (k))) \\ Rémy Sigrist, Dec 08 2019

More terms from Rémy Sigrist, Dec 08 2019

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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