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

%I A343193 #39 Jun 13 2021 13:35:36
%S A343193 1,9279,92434863,923988964495,9239427676877311,92393887177379735327,
%T A343193 923938441006918271400831,9239384074081430755652624559,
%U A343193 92393840333765561759423951663423,923938402972369921481535120722882015
%N A343193 Number of ordered quadruples (w, x, y, z) with gcd(w, x, y, z) = 1 and 1 <= {w, x, y, z} <= 10^n.
%D A343193 Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.
%H A343193 Chai Wah Wu, <a href="/A343193/b343193.txt">Table of n, a(n) for n = 0..15</a>
%F A343193 Lim_{n->infinity} a(n)/10^(4*n) = 1/zeta(4) = A215267 = 90/Pi^4.
%F A343193 a(n) = A082540(10^n).
%e A343193 (1,2,2,3) is counted, but (2,4,4,6) is not, because gcd = 2.
%e A343193 For n=1, the size of the division tesseract matrix is 10 X 10 X 10 X 10:
%e A343193 .
%e A343193               o------------x(w=10)------------o
%e A343193              /|.                            ./ |
%e A343193             / |.                           ./  |
%e A343193            /  |.                          ./   |
%e A343193           /   |.                         ./    |
%e A343193          /    |.                      z(w=10)  |
%e A343193         /     |.                      . /      |
%e A343193        /      |.                     . /       |
%e A343193       /       |.                   .  /     y(w=10)
%e A343193      o------------------------------.o         |
%e A343193     |\        /|¯¯¯¯¯¯x(w=1)¯¯¯¯¯¯/. |         |
%e A343193     | w      / |                 /.| |         |
%e A343193     |  \ z(w=1)|                /. | |         |
%e A343193     |   \  /   |y(w=1)         /.  | |         |
%e A343193     |    \/-------------------/.   | |         |
%e A343193     |     |                   |    | |         |        w | sums
%e A343193     |     |  Cube at w = 1    |    | |         |      ----+-----
%e A343193     |     |   10 X 10 X 10    | _ _| |---------o        1 | 1000
%e A343193     |     |    contains       |    / |         /        2 |  875
%e A343193     |     |      1000         |   /  |        /         3 |  973
%e A343193     |     |    completely     |  /   |       /          4 |  875
%e A343193     |     | reduced fractions | /    |      /           5 |  992
%e A343193     |     |                   |/     |     /            6 |  849
%e A343193     |     /------------------- \     |    /             7 |  999
%e A343193     |    /                      \    |   /              8 |  875
%e A343193     |   w                        w   |  /               9 |  973
%e A343193     |  /                          \  | /               10 |  868
%e A343193     | /                            \ |/               ----+-----
%e A343193     o -------------------------------o       sum for a(1) | 9279
%o A343193 (Python)
%o A343193 from labmath import mobius
%o A343193 def A343193(n): return sum(mobius(k)*(10**n//k)**4 for k in range(1, 10**n+1))
%Y A343193 Cf. A215267, A013662, A082540, A342841 (3D), A342586 (2D).
%Y A343193 Related counts of k-tuples:
%Y A343193 pairs: A018805, A342632, A342586;
%Y A343193 triples: A071778, A342935, A342841;
%Y A343193 quadruples: A082540, A343527, A343193;
%Y A343193 5-tuples: A343282;
%Y A343193 6-tuples: A343978, A344038. - _N. J. A. Sloane_, Jun 13 2021
%K A343193 nonn
%O A343193 0,2
%A A343193 _Karl-Heinz Hofmann_, Apr 07 2021
%E A343193 Edited by _N. J. A. Sloane_, Jun 13 2021