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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.
7
1, 15, 239, 3823, 60735, 972191, 15517679, 248252879, 3969108895, 63506982943, 1015951568815, 16255093526239, 260068569617727, 4161109496115135, 66577084386669199, 1065232436999055375, 17043668344393625999, 272698739815301095247, 4363176901343767529551, 69810828455823683068415, 1116973047989955380768527
OFFSET
0,2
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..52 (n = 0..31 from Karl-Heinz Hofmann)
FORMULA
Lim_{n->infinity} a(n)/2^(4*n) = 1/zeta(4) = A215267 = 90/Pi^4.
a(n) = A082540(2^n).
EXAMPLE
.
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
PROG
(Python)
from labmath import mobius
def A343527(n): return sum(mobius(k)*(2**n//k)**4 for k in range(1, 2**n+1))
KEYWORD
nonn
AUTHOR
Karl-Heinz Hofmann, Apr 18 2021
EXTENSIONS
Edited by N. J. A. Sloane, Jun 13 2021
STATUS
approved