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A342841 Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 10^n. 8
1, 841, 832693, 832046137, 831916552903, 831908477106883, 831907430687799769, 831907383078281024371, 831907373418800027750413, 831907372722449100147414487, 831907372589073124899487831735, 831907372581823023465031521920149, 831907372580768386561159867257319711 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

FORMULA

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

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

EXAMPLE

For visualisation, 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)

PROG

(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)

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

Sequence in context: A253599 A121499 A253514 * A337730 A049530 A158404

Adjacent sequences:  A342838 A342839 A342840 * A342842 A342843 A342844

KEYWORD

nonn,hard

AUTHOR

Karl-Heinz Hofmann, Mar 24 2021

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

STATUS

approved

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Last modified August 3 03:52 EDT 2021. Contains 346435 sequences. (Running on oeis4.)