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

%I A343527 #24 Jun 13 2021 13:21:44
%S A343527 1,15,239,3823,60735,972191,15517679,248252879,3969108895,63506982943,
%T A343527 1015951568815,16255093526239,260068569617727,4161109496115135,
%U A343527 66577084386669199,1065232436999055375,17043668344393625999,272698739815301095247,4363176901343767529551,69810828455823683068415,1116973047989955380768527
%N A343527 Number of ordered quadruples (w, x, y, z) with gcd(w, x, y, z) = 1 and 1 <= {w, x, y, z} <= 2^n.
%H A343527 Chai Wah Wu, <a href="/A343527/b343527.txt">Table of n, a(n) for n = 0..52</a> (n = 0..31 from Karl-Heinz Hofmann)
%F A343527 Lim_{n->infinity} a(n)/2^(4*n) = 1/zeta(4) = A215267 = 90/Pi^4.
%F A343527 a(n) = A082540(2^n).
%e A343527 .
%e A343527 For n=3, the size of the gris is 8 X 8 X 8 X 8:
%e A343527 .
%e A343527               o------------x(w=8)-------------o
%e A343527              /|.                            ./ |
%e A343527             / |.                           ./  |
%e A343527            /  |.                          ./   |
%e A343527           /   |.                         ./    |
%e A343527          /    |.                      z(w=8)   |
%e A343527         /     |.                      . /      |
%e A343527        /      |.                     . /       |
%e A343527       /       |.                   .  /     y(w=8)
%e A343527      o------------------------------.o         |
%e A343527     |\        /|¯¯¯¯¯¯x(w=1)¯¯¯¯¯¯/. |         |
%e A343527     | w      / |                 /.| |         |
%e A343527     |  \ z(w=1)|                /. | |         |
%e A343527     |   \  /   |y(w=1)         /.  | |         |
%e A343527     |    \/-------------------/.   | |         |
%e A343527     |     |                   |    | |         |        w | sums
%e A343527     |     |  Cube at w = 1    |    | |         |      ----+-----
%e A343527     |     |    8 X 8 X 8      | _ _| |---------o        1 |  512
%e A343527     |     |    contains       |    / |         /        2 |  448
%e A343527     |     |       512         |   /  |        /         3 |  504
%e A343527     |     |    completely     |  /   |       /          4 |  448
%e A343527     |     | reduced fractions | /    |      /           5 |  511
%e A343527     |     |                   |/     |     /            6 |  441
%e A343527     |     /------------------- \     |    /             7 |  511
%e A343527     |    /                      \    |   /              8 |  448
%e A343527     |   w                        w   |  /             ----+-----
%e A343527     |  /                          \  | /     sum for a(3) | 3823
%e A343527     | /                            \ |/
%e A343527     o -------------------------------o
%o A343527 (Python)
%o A343527 from labmath import mobius
%o A343527 def A343527(n): return sum(mobius(k)*(2**n//k)**4 for k in range(1, 2**n+1))
%Y A343527 Cf. A018805, A342632, A342586, A071778.
%Y A343527 Cf. A342935, A342841, A082540, A343193.
%K A343527 nonn
%O A343527 0,2
%A A343527 _Karl-Heinz Hofmann_, Apr 18 2021
%E A343527 Edited by _N. J. A. Sloane_, Jun 13 2021