A342935 Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 2^n.

1, 7, 55, 439, 3433, 27541, 218773, 1749223, 13964245, 111725197, 893433661, 7147232467, 57169672861, 457364647435, 3658819119307, 29270432746633, 234161501271463, 1873293863661469, 14986321908515773, 119890565631185995, 959124025074311215, 7672992332048493361

Offset: 0

Author

Karl-Heinz Hofmann, Mar 29 2021

Keywords

• nonn

Examples

			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)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..53 (terms n = 0..32 from Karl-Heinz Hofmann)

Crossrefs

Cf. A088453, A002117, A018805, A342632, A342586, A071778, A342841.

Programs

Mathematica

Array[Sum[MoebiusMu[k]*Floor[(2^#)/k]^3, {k, 2^# + 1}] &, 22, 0] (* Michael De Vlieger, Apr 05 2021 *)

Python

from labmath import mobius
def A342935(n): return sum(mobius(k)*(2**n//k)**3 for k in range(1, 2**n+1))

Formula

Lim_{n->infinity} a(n)/2^(3*n) = 1/zeta(3) = A088453 = 1/Apéry's constant.

a(n) = A071778(2^n).

Extensions

Edited by N. J. A. Sloane, Jun 13 2021