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

0,2

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

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

Example

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).

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}]

Prog

(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

Crossrefs

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

Sequence in context: A356377 A075380 A349374 * A248497 A255439 A177983

Adjacent sequences: A382862 A382863 A382864 * A382866 A382867 A382868

Keyword

nonn,look

Author

Federico Provvedi, May 21 2025

Status

approved