A343193 Number of ordered quadruples (w, x, y, z) with gcd(w, x, y, z) = 1 and 1 <= {w, x, y, z} <= 10^n.

1, 9279, 92434863, 923988964495, 9239427676877311, 92393887177379735327, 923938441006918271400831, 9239384074081430755652624559, 92393840333765561759423951663423, 923938402972369921481535120722882015

Offset: 0

Author

Karl-Heinz Hofmann, Apr 07 2021

Keywords

• nonn

Examples

			(1,2,2,3) is counted, but (2,4,4,6) is not, because gcd = 2.
For n=1, the size of the division tesseract matrix is 10 X 10 X 10 X 10:
.
              o------------x(w=10)------------o
             /|.                            ./ |
            / |.                           ./  |
           /  |.                          ./   |
          /   |.                         ./    |
         /    |.                      z(w=10)  |
        /     |.                      . /      |
       /      |.                     . /       |
      /       |.                   .  /     y(w=10)
     o------------------------------.o         |
    |\        /|¯¯¯¯¯¯x(w=1)¯¯¯¯¯¯/. |         |
    | w      / |                 /.| |         |
    |  \ z(w=1)|                /. | |         |
    |   \  /   |y(w=1)         /.  | |         |
    |    \/-------------------/.   | |         |
    |     |                   |    | |         |        w | sums
    |     |  Cube at w = 1    |    | |         |      ----+-----
    |     |   10 X 10 X 10    | _ _| |---------o        1 | 1000
    |     |    contains       |    / |         /        2 |  875
    |     |      1000         |   /  |        /         3 |  973
    |     |    completely     |  /   |       /          4 |  875
    |     | reduced fractions | /    |      /           5 |  992
    |     |                   |/     |     /            6 |  849
    |     /------------------- \     |    /             7 |  999
    |    /                      \    |   /              8 |  875
    |   w                        w   |  /               9 |  973
    |  /                          \  | /               10 |  868
    | /                            \ |/               ----+-----
    o -------------------------------o       sum for a(1) | 9279
		

References

• Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..15

Crossrefs

Cf. A215267, A013662, A082540, A342841 (3D), A342586 (2D).

Related counts of k-tuples:

pairs: A018805, A342632, A342586;

triples: A071778, A342935, A342841;

quadruples: A082540, A343527, A343193;

5-tuples: A343282;

6-tuples: A343978, A344038. - N. J. A. Sloane, Jun 13 2021

Programs

Python

from labmath import mobius
def A343193(n): return sum(mobius(k)*(10**n//k)**4 for k in range(1, 10**n+1))

Formula

Lim_{n->infinity} a(n)/10^(4*n) = 1/zeta(4) = A215267 = 90/Pi^4.

a(n) = A082540(10^n).

Extensions

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