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A342935
Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 2^n.
8
1, 7, 55, 439, 3433, 27541, 218773, 1749223, 13964245, 111725197, 893433661, 7147232467, 57169672861, 457364647435, 3658819119307, 29270432746633, 234161501271463, 1873293863661469, 14986321908515773, 119890565631185995, 959124025074311215, 7672992332048493361
OFFSET
0,2
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..53 (terms n = 0..32 from Karl-Heinz Hofmann)
FORMULA
Lim_{n->infinity} a(n)/2^(3*n) = 1/zeta(3) = A088453 = 1/Apéry's constant.
a(n) = A071778(2^n).
EXAMPLE
For n=3, the size of the division cube matrix is 8 X 8 X 8:
.
: : : : : : : : :
.
z = 4 | 1 2 3 4 5 6 7 8
------+----------------------
1 /| o o o o o o o o 8
2 / | o . o . o . o . 4 64 Sum (z = 1)
3/ | o o o o o o o o 8 /
/ o . 4 48 Sum (z = 2)
z = 5 |/1 2 3 4 5 6 7 8 o 8 /
------+---------------------- 4 60 Sum (z = 3)
1 /| o o o o o o o o 8 8 /
2 / | o o o o o o o o 8 4 /
3/ | o o o o o o o o 8 --/
/ o o 8 48 Sum (z = 4)
z = 6 |/1 2 3 4 5 6 7 8 o 7 /
------+---------------------- 8 /
1 /| o o o o o o o o 8 8 /
2 / | o . o . o . o . 4 8 /
3/ | o o o o o o o o 6 --/
/ o . 4 63 Sum (z = 5)
z = 7 |/1 2 3 4 5 6 7 8 o 8 /
------+---------------------- 3 /
1 /| o o o o o o o o 8 8 /
2 / | o o o o o o o o 8 4 /
3/ | o o o o o o o o 8 --/
/ o o 8 45 Sum (z = 6)
z = 8 |/1 2 3 4 5 6 7 8 o 8 /
------+---------------------- 8 /
1 | o o o o o o o o 8 7 /
2 | o . o . o . o . 4 8 /
3 | o o o o o o o o 8 --/
4 | o . o . o . o . 4 63 Sum (z = 7)
5 | o o o o o o o o 8 /
6 | o . o . o . o . 4 /
7 | o o o o o o o o 8 /
8 | o . o . o . o . 4 /
--/
48 Sum (z = 8)
|
---
439 Cube Sum (z = 1..8)
MATHEMATICA
Array[Sum[MoebiusMu[k]*Floor[(2^#)/k]^3, {k, 2^# + 1}] &, 22, 0] (* Michael De Vlieger, Apr 05 2021 *)
PROG
(Python)
from labmath import mobius
def A342935(n): return sum(mobius(k)*(2**n//k)**3 for k in range(1, 2**n+1))
KEYWORD
nonn
AUTHOR
Karl-Heinz Hofmann, Mar 29 2021
EXTENSIONS
Edited by N. J. A. Sloane, Jun 13 2021
STATUS
approved