A382865 Bitwise XOR of all integers between n and 2n (endpoints included).

0, 3, 5, 4, 8, 15, 13, 8, 16, 27, 21, 28, 24, 23, 29, 16, 32, 51, 37, 52, 40, 63, 45, 56, 48, 43, 53, 44, 56, 39, 61, 32, 64, 99, 69, 100, 72, 111, 77, 104, 80, 123, 85, 124, 88, 119, 93, 112, 96, 83, 101, 84, 104, 95, 109, 88, 112, 75, 117, 76, 120, 71, 125, 64, 128, 195

Offset: 0

Author

Federico Provvedi, May 21 2025

Keywords

• nonn
• look

Examples

			a(3) = 3 XOR 4 XOR 5 XOR 6 = 4, in binary representation is: ((011 XOR 100) XOR 101) XOR 110 = (111 XOR 101) XOR 110 = 010 XOR 110 = 100 (4 in decimal).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..16383
• Karl-Heinz Hofmann, Zoom trip with n mod 4 colored.
• Wikipedia, Bitwise operation: XOR

Crossrefs

Cf. A038712, A047615, A065621, A114389, A010873, A048724, A174091, A181983.

Programs

Maple

a:= proc(n) option remember; uses Bits; `if`(n=0, 0,
      Xor(Xor(Xor(a(n-1), n-1), 2*n-1), 2*n))
    end:
seq(a(n), n=0..65);  # Alois P. Heinz, May 26 2025

Mathematica

a[n_] = BitXor[BitOr[n-1, 2] - (-1)^n*(n-1), 4*n]/2; Table[a[n],{n,0, 65}]

PARI

a(n) = my(b=n); for (i=n+1, 2*n, b = bitxor(b, i)); b; \\ Michel Marcus, May 25 2025

Python

def A382865(n): return [0, n, 1, n-1][n%4] ^ (2*n) # Karl-Heinz Hofmann, May 26 2025

Formula

a(2n) = A047615(n+1), and for every integer k>1: a(n*2^k -1) = 2^k * A065621(n).

a(4n) = 8*n, a(4n+1) = 2*A114389(n+1) + 1, a(4n+2) = 8*n + 5, a(4n+3) = 4*A065621(n+1).

From Karl-Heinz Hofmann, May 27 2025: (Start)

For all n == 0 (mod 4) --> a(n) = A005843(n) = 2*n

For all n == 1 (mod 4) --> a(n) = A048724(n)

For all n == 2 (mod 4) --> a(n) = A005408(n) = 2*n + 1

For all n == 3 (mod 4) --> a(n) = A048724(n) - 1 (End)

a(n) = (1/2) * XOR(A174091(n-1) - A181983(n-1), 4*n). - Federico Provvedi, May 31 2025