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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.)