cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

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%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