A342841 Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 10^n.

1, 841, 832693, 832046137, 831916552903, 831908477106883, 831907430687799769, 831907383078281024371, 831907373418800027750413, 831907372722449100147414487, 831907372589073124899487831735, 831907372581823023465031521920149, 831907372580768386561159867257319711

Offset: 0

Author

Karl-Heinz Hofmann, Mar 24 2021

Keywords

• nonn
• hard

Examples

			For visualization, the set(x, y, z) is inscribed in a cube matrix.
"o" stands for a gcd = 1.
"." stands for a gcd > 1.
.
For n=1, the size of the cube matrix is 10 X 10 X 10:
.
                         / : : : : : : : : : :
                        /                               100 Sum (z = 1)
                z = 7 |/1 2 3 4 5 6 7 8 9 10             |
                    --+---------------------             75 Sum (z = 2)
                   1 /| o o o o o o o o o o    10        |
                   2/ | o o o o o o o o o o    10        91 Sum (z = 3)
                   /                           10        |
           z = 8 |/1 2 3 4 5 6 7 8 9 10        10       75 Sum (z = 4)
               --+---------------------        10      /
              1 /| o o o o o o o o o o    10   10     96 Sum (z = 5)
              2/ | o . o . o . o . o .     5    9    /
              /                           10   10   67 Sum (z = 6)
      z = 9 |/1 2 3 4 5 6 7 8 9 10         5   10  /
          --+---------------------        10   10 /
         1 /| o o o o o o o o o o    10    5   --/
         2/ | o o o o o o o o o o    10   10   99 Sum (z = 7)
         /                            7    5   /
z = 10 |/1 2 3 4 5 6 7 8 9 10        10   10  /
     --+---------------------        10    5 /
     1 | o o o o o o o o o o    10    7   --/
     2 | o . o . o . o . o .     5   10   75 Sum (z = 8)
     3 | o o o o o o o o o o    10   10   /
     4 | o . o . o . o . o .     5    7  /
     5 | o o o o . o o o o .     8   10 /
     6 | o . o . o . o . o .     5   --/
     7 | o o o o o o o o o o    10   91 Sum (z = 9)
     8 | o . o . o . o . o .     5   /
     9 | o o o o o o o o o o    10  /
    10 | o . o . . . o . o .     4 /
                                --/
                                72 Sum (z = 10)
                                /
                               |
                            ------
                              841 Cube Sum (z = 1..10)
		

References

• Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..15
• Karl-Heinz Hofmann, An animation of the cube with n = 1.

Crossrefs

Cf. A342586 (for 10^n X 10^n), A018805, A002117 (zeta(3)), A071778.

Related counts of k-tuples:

pairs: A018805, A342632, A342586;

triples: A071778, A342935, A342841;

quadruples: A082540, A343527, A343193;

5-tuples: A343282;

6-tuples: A343978, A344038. - N. J. A. Sloane, Jun 13 2021

Programs

Python

import math
for n in range (0, 10):
     counter = 0
     for x in range (1, pow(10, n)+1):
        for y in range(1, pow(10, n)+1):
            for z in range(1, pow(10, n)+1):
                if math.gcd(math.gcd(y, x),z) ==  1:
                    counter += 1
     print(n, counter)

Formula

Lim_{n->infinity} a(n)/10^(3*n) = 1/zeta(3) = 1/Apéry's constant.

a(n) = A071778(10^n).

Extensions

a(5)-a(10) from Hugo Pfoertner, Mar 25 2021

a(11) from Hugo Pfoertner, Mar 26 2021

a(12) from Bruce Garner, Mar 29 2021