A343527 Number of ordered quadruples (w, x, y, z) with gcd(w, x, y, z) = 1 and 1 <= {w, x, y, z} <= 2^n.

1, 15, 239, 3823, 60735, 972191, 15517679, 248252879, 3969108895, 63506982943, 1015951568815, 16255093526239, 260068569617727, 4161109496115135, 66577084386669199, 1065232436999055375, 17043668344393625999, 272698739815301095247, 4363176901343767529551, 69810828455823683068415, 1116973047989955380768527

Offset: 0

Author

Karl-Heinz Hofmann, Apr 18 2021

Keywords

• nonn

Examples

			.
For n=3, the size of the gris is 8 X 8 X 8 X 8:
.
              o------------x(w=8)-------------o
             /|.                            ./ |
            / |.                           ./  |
           /  |.                          ./   |
          /   |.                         ./    |
         /    |.                      z(w=8)   |
        /     |.                      . /      |
       /      |.                     . /       |
      /       |.                   .  /     y(w=8)
     o------------------------------.o         |
    |\        /|¯¯¯¯¯¯x(w=1)¯¯¯¯¯¯/. |         |
    | w      / |                 /.| |         |
    |  \ z(w=1)|                /. | |         |
    |   \  /   |y(w=1)         /.  | |         |
    |    \/-------------------/.   | |         |
    |     |                   |    | |         |        w | sums
    |     |  Cube at w = 1    |    | |         |      ----+-----
    |     |    8 X 8 X 8      | _ _| |---------o        1 |  512
    |     |    contains       |    / |         /        2 |  448
    |     |       512         |   /  |        /         3 |  504
    |     |    completely     |  /   |       /          4 |  448
    |     | reduced fractions | /    |      /           5 |  511
    |     |                   |/     |     /            6 |  441
    |     /------------------- \     |    /             7 |  511
    |    /                      \    |   /              8 |  448
    |   w                        w   |  /             ----+-----
    |  /                          \  | /     sum for a(3) | 3823
    | /                            \ |/
    o -------------------------------o
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..52 (n = 0..31 from Karl-Heinz Hofmann)

Crossrefs

Cf. A018805, A342632, A342586, A071778.

Cf. A342935, A342841, A082540, A343193.

Programs

Python

from labmath import mobius
def A343527(n): return sum(mobius(k)*(2**n//k)**4 for k in range(1, 2**n+1))

Formula

Lim_{n->infinity} a(n)/2^(4*n) = 1/zeta(4) = A215267 = 90/Pi^4.

a(n) = A082540(2^n).

Extensions

Edited by N. J. A. Sloane, Jun 13 2021