cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

This page as a plain text file.
%I A342841 #76 Jan 06 2025 21:06:15
%S A342841 1,841,832693,832046137,831916552903,831908477106883,
%T A342841 831907430687799769,831907383078281024371,831907373418800027750413,
%U A342841 831907372722449100147414487,831907372589073124899487831735,831907372581823023465031521920149,831907372580768386561159867257319711
%N A342841 Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 10^n.
%D A342841 Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.
%H A342841 Chai Wah Wu, <a href="/A342841/b342841.txt">Table of n, a(n) for n = 0..15</a>
%H A342841 Karl-Heinz Hofmann, <a href="/A342841/a342841.gif">An animation of the cube with n = 1</a>.
%F A342841 Lim_{n->infinity} a(n)/10^(3*n) = 1/zeta(3) = 1/Apéry's constant.
%F A342841 a(n) = A071778(10^n).
%e A342841 For visualization, the set(x, y, z) is inscribed in a cube matrix.
%e A342841 "o" stands for a gcd = 1.
%e A342841 "." stands for a gcd > 1.
%e A342841 .
%e A342841 For n=1, the size of the cube matrix is 10 X 10 X 10:
%e A342841 .
%e A342841                          / : : : : : : : : : :
%e A342841                         /                               100 Sum (z = 1)
%e A342841                 z = 7 |/1 2 3 4 5 6 7 8 9 10             |
%e A342841                     --+---------------------             75 Sum (z = 2)
%e A342841                    1 /| o o o o o o o o o o    10        |
%e A342841                    2/ | o o o o o o o o o o    10        91 Sum (z = 3)
%e A342841                    /                           10        |
%e A342841            z = 8 |/1 2 3 4 5 6 7 8 9 10        10       75 Sum (z = 4)
%e A342841                --+---------------------        10      /
%e A342841               1 /| o o o o o o o o o o    10   10     96 Sum (z = 5)
%e A342841               2/ | o . o . o . o . o .     5    9    /
%e A342841               /                           10   10   67 Sum (z = 6)
%e A342841       z = 9 |/1 2 3 4 5 6 7 8 9 10         5   10  /
%e A342841           --+---------------------        10   10 /
%e A342841          1 /| o o o o o o o o o o    10    5   --/
%e A342841          2/ | o o o o o o o o o o    10   10   99 Sum (z = 7)
%e A342841          /                            7    5   /
%e A342841 z = 10 |/1 2 3 4 5 6 7 8 9 10        10   10  /
%e A342841      --+---------------------        10    5 /
%e A342841      1 | o o o o o o o o o o    10    7   --/
%e A342841      2 | o . o . o . o . o .     5   10   75 Sum (z = 8)
%e A342841      3 | o o o o o o o o o o    10   10   /
%e A342841      4 | o . o . o . o . o .     5    7  /
%e A342841      5 | o o o o . o o o o .     8   10 /
%e A342841      6 | o . o . o . o . o .     5   --/
%e A342841      7 | o o o o o o o o o o    10   91 Sum (z = 9)
%e A342841      8 | o . o . o . o . o .     5   /
%e A342841      9 | o o o o o o o o o o    10  /
%e A342841     10 | o . o . . . o . o .     4 /
%e A342841                                 --/
%e A342841                                 72 Sum (z = 10)
%e A342841                                 /
%e A342841                                |
%e A342841                             ------
%e A342841                               841 Cube Sum (z = 1..10)
%o A342841 (Python)
%o A342841 import math
%o A342841 for n in range (0, 10):
%o A342841      counter = 0
%o A342841      for x in range (1, pow(10, n)+1):
%o A342841         for y in range(1, pow(10, n)+1):
%o A342841             for z in range(1, pow(10, n)+1):
%o A342841                 if math.gcd(math.gcd(y, x),z) ==  1:
%o A342841                     counter += 1
%o A342841      print(n, counter)
%Y A342841 Cf. A342586 (for 10^n X 10^n), A018805, A002117 (zeta(3)), A071778.
%Y A342841 Related counts of k-tuples:
%Y A342841 pairs: A018805, A342632, A342586;
%Y A342841 triples: A071778, A342935, A342841;
%Y A342841 quadruples: A082540, A343527, A343193;
%Y A342841 5-tuples: A343282;
%Y A342841 6-tuples: A343978, A344038. - _N. J. A. Sloane_, Jun 13 2021
%K A342841 nonn,hard
%O A342841 0,2
%A A342841 _Karl-Heinz Hofmann_, Mar 24 2021
%E A342841 a(5)-a(10) from _Hugo Pfoertner_, Mar 25 2021
%E A342841 a(11) from _Hugo Pfoertner_, Mar 26 2021
%E A342841 a(12) from _Bruce Garner_, Mar 29 2021